International Journal of Modern Physics C, fc World Scientic Publishing Company AN EXACT FINITE FIELD RENORMALIZATION GROUP CALCULATION ON A TWO{DIMEN

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1 International Journal of Modern Physics C, fc World Scientic Publishing Company AN EXACT FINITE FIELD RENORMALIZATION GROUP CALCULATION ON A TWO{DIMENSIONAL FRACTAL LATTICE J.F. Nystrom Department of Electrical and Computer Engineering, PO Box 4423 University of Idaho, Moscow, ID , USA nystrom(at)ee.uidaho.edu Received (received date) Revised (revised date) A two{dimensional fractal lattice, herein referred to as the tetrahedral gasket, is used as a model for an exact two{dimensional real-space renormalization group calculation. It is shown that this Ising spin{system has exact solutions which includes a non{trivial phase transition even in the presence of a nite eld. The calculation also introduces a new analysis tool, the free energy surface plot, which gives further insight into the phase diagram of the spin{system. The discussion includes comments concerning the apparent preference of the system to maintain some nite entropy, even in the presence of an extremely large spin{spin coupling. Keywords: Real-space Renormalization Group, Ising Model, Fractal Lattice.. Introduction A Real-Space Renormalization Group (RSRG) calculation is performed on the tetrahedral gasket []. Three levels of the tetrahedral gasket are shown in Figure. It is fairly common to apply the RSRG to fractal geometries [2, 3, 4, 5, 6, 7]. The model presented herein is an extension to the Sierpinski gasket with fractal periodic boundary conditions [8]. The type of results that are expected from a RSRG analysis include a phase diagram and the characterization of the model's critical behavior [2, 3, 4, 9], which along with the model's partition function, completely describe the thermodynamics of the system. The critical behavior of a system is well characterized by the location of critical points and the values of the critical exponents [, ]. The calculation technique contained herein is patterned after [2, 3, 4]. The tetrahedral gasket has a (fractal) dimension of exactly two, and the RSRG calculation on the tetrahedral gasket is an exact calculation when both even and odd interactions are present. Thus, the tetrahedral gasket is a two-dimensional model with an exact solution in the presence of a nite eld. In this report, the geometry and dimensionality of the tetrahedral gasket are introduced, and the RSRG calculation is derived for the tetrahedral gasket. This derivation (of the RSRG cal-

2 2 Exact two{dimensional renormalization group calculation Figure : Three levels of the tetrahedral gasket. culation) includes the development of the Hamiltonians, partition functions and the (reduced) free energy. The results include a detailed analysis of the model's phase diagram and an introduction to the free energy surface plot. These surface plots evidently have features that correspond to attributes exhibited on the phase diagram. The critical behavior of the tetrahedral gasket is calculated, where the technique of linearizing near the critical point is used to calculate the model's critical exponents. The critical exponent is also calculated numerically from rst principles. 2. Geometry This section provides an introduction to the geometry of the RSRG model and discusses the dimensionality of the tetrahedral gasket. The model presented is built as an extension to the Sierpinski gasket [3, 4, 8]. The motivation that led to the construction of the tetrahedral gasket from the base tetrahedron is attributable to the author's familiarity with the geometrical constructions of R. Buckminster Fuller [5]. In [6], the tetrahedral gasket is referred to as a Sierpinski tetrahedron, while in [7], Mandelbrot calls this fractal lattice a fractal skewed web. 2.. Dimensionality of the tetrahedral gasket For the tetrahedral gasket, we start with a base tetrahedron. We remove an interior octahedron from this structure and are left with four tetrahedrons situated within the four corners of the original structure. The procedure is then repeated again on

3 Two{dimensional fractal 3 each of the four tetrahedrons to give sixteen tetrahedrons. The rst three levels of this procedure is shown in Figure. 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. The relationship between the new volume V and the original volume V is then given by: V = b d V : () 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 : (2) For the tetrahedral gasket, when the octahedral segment is removed from the tetrahedron, this could be viewed as adding three more tetrahedrons to our base one, in a sense doubling the length of the side of the original tetrahedron, except now it has a void in the middle. Thus, with V gasket = 4 V gasket ; the dimensionality of the tetrahedral gasket is d gasket = 2 : Therefore the tetrahedral gasket is a full two-dimensional object. Apparently, the octahedral void of the tetrahedral gasket must account for four tetrahedral volumes Coordination numbers The coordination number [9], i.e., the number of nearest-neighbor sites, is one method of characterizing a geometry. For the tetrahedral gasket, the interior nodes have a coordination number dierent from the nodes that occupy the four corners of the gasket (see Figure 2). But note that each corner node is shared between two separate congurations (e.g., when constructing the next level of the gasket). When this sharing of nodes is taken into account, the eective coordination number for all nodes of the tetrahedral gasket is six. 3. Renormalization group calculation This section develops the RSRG calculations for the tetrahedral gasket. The Hamiltonian and associated recurrence relations are specied separately for the model with even interactions only, and then for the model with both even and odd interactions. An interesting feature of the tetrahedral gasket is that it has dimensionality of exactly two and exact recurrence relations can be calculated with both even and odd interactions. Although the critical point for the tetrahedral gasket is not calculated exactly (but rather found from numerical considerations), the partition functions for

4 4 Exact two{dimensional renormalization group calculation S S6 S7 S5 S2 S S9 S4 S8 S3 Figure 2: Tetrahedral decimation. the tetrahedral gasket arrangement in Figure 2 and the decimated tetrahedron can be equated to provide exact recurrence relations. All calculations were performed using the Mathematica[2] programming environment. 3.. The decimation procedure Consider the arrangement in Figure 2, which can be viewed as the state after the rst octahedral segment is removed from a base tetrahedron. The four tetrahedrons that were produced are the volumes 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 : The octahedral void is enclosed by the six spins labeled with dots: fs5; s6; s7; s8; s9; sg : The decimation consists of summing over all congurations of the six interior spins (s i = +;?; 8i; 5 i ), and 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 the 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. If, during the renormalization procedure, new types of couplings are not generated we say the group (of couplings) is closed. If the group of couplings

5 Two{dimensional fractal 5 is closed without any approximations or simplications to the geometry, we say the calculations are exact. The recurrence relations for the tetrahedral gasket are exact when either even, or both even and odd couplings are used. Note: the decimation works on all levels. If we decimate each of the four tetrahedrons with octahedral voids (four of them) in the third level of the gasket (in Figure ), the second level gasket is produced. Applied again, the base tetrahedron results. The subtle assumption that is made when performing the RSRG calculations is that we start with a n th level gasket, with n, such that the renormalization can be performed as many times as needed to approach asymptotic values for the couplings Hamiltonian for even interactions The details of the model are presented by specifying the couplings used in the Hamiltonian and by describing the development of the recurrence relations. For the system displayed in Figure 2, 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. The k and m couplings are more straightforward than the l interaction. The basic Sierpinski gasket model was extended [8] to include the l coupling, which in a sense provides for fractal periodic boundary conditions [8]. When the interactions k, l, and m are used, the Hamiltonian for the system in Figure 2 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 +s4s5 + s4s9 + s3s7 + s3s + s3s6) + m (ss5s6s7 + s2s6s9s + s3s5s8s9 + s4s7s8s) : (3) Compare (3) to the Hamiltonian for the base tetrahedron (spins fs; s2; s3; s4g only), which is given by?h = 2 k + k (ss2 + ss3 + ss4 + s2s3 + s2s4 + s3s4) + m (ss2s3s4) : (4) The zero spin coupling, k, contributes to the reduced free energy (see [2] and (4)). For the tetrahedral gasket, the outside spins in Figure 2 are shared between The most popular approximation is a bond-moving method, the so{called Migdal-Kadano approximation ([2, 2, 3] and references therein).

6 6 Exact two{dimensional renormalization group calculation Z ss2s3s4 s s2 s3 s4 Z pppp Z pppm Z ppmm Table : Labeling of partial partition functions. two separate decimations, therefore the four exterior spins each contribute a =2 k term to the Hamiltonian for the base tetrahedron, as shown in (4). The partition function for the system in Figure 2 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 (5) 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 The renormalization-group calculation requires that the partition function be preserved (Z N = Z N ). Thus we equate (5) and (6), which relates the primed and original couplings: (6) Tr Tr e?h = Tr e?h. fs; s2; s3; s4g fs5; s6; s7; s8; s9; sg fs; s2; s3; s4g (7) Now, calculate the recurrence relations between the interaction couplings given in (3) and (4), which give the functions: k (k; l; m); k (k; l; m); and m (k; l; m) : (8) In order to describe completely the functions in (8), consider the partial partition function, Z ss2s3s4, where the conguration of spins fs; s2; s3; s4g replace their respective position in the partial partition function as shown in Table. It can be veried (using (6)), that the following relations hold: Z pppp = Z mmmm ; Z pppm = Z ppmp = Z pmpp = Z mppp = Z mmmp = Z mmpm = Z mpmm = Z pmmm ; Z ppmm = Z pmmp = Z mmpp = Z pmpm = Z mppm = Z mpmp : Therefore, out of 6 possible partition functions given by the evaluation of (6), only three are independent. These three equations represent the need for three couplings for the base tetrahedron. Evidently the second nearest-neighbor interaction parameter l does not get renormalized, since it does not appear in the Hamiltonian for the

7 Two{dimensional fractal 7 base tetrahedron. Thus, the l coupling has a constant value throughout the renormalization calculations. Evaluation of (6) then gives the following simultaneous equations in the renormalized couplings: ln[z pppp ] = 2k + 6k + m ln[z pppm ] = 2k? m and solving (9), the functions in (8) are given by: ln[z ppmm ] = 2k? 2k + m ; (9) k (k; l; m) = (=6)(ln[Z pppp ] + 3 ln[z ppmm ] + 4 ln[z pppm ]) k (k; l; m) = (=8)(ln[Z pppp ]? ln[z ppmm ]) m (k; l; m) = (=8)(ln[Z pppp ] + 3 ln[z ppmm ]? 4 ln[z pppm ]) : () Note that we now need values for the partial partition function, Z ss2s3s4 (k; l; m). These functions are derived by using (5) such that Z ss2s3s4 = Tr e?h, fs5; s6; s7; s8; s9; sg () which give the following partial partition functions: Z pppp (k; l; m) = 2e?4k + 6e?(4k+8l) + 2e?(4k+4l) + 2e (4k+4l) + 6e (2k+8l) +e?(2l+4m) + 3e (4l?4m) + 4e?4m + 4e 4m + 3e?(8k+4l?4m) +e (24k+2l+4m) ; Z pppm (k; l; m) = 2e (6k?6l?2m) + 5e?(2k+2l+2m) + 3e (6k?2l?2m) +6e?(2k?2l+2m) + 3e?(2k?6l+2m) + 3e (6k+6l?2m) +6e (2k?6l+2m) + 3e?(6k+2l?2m) + 3e (2k?2l+2m) +9e?(6k?2l?2m) + 6e (2k+2l+2m) + 3 (k+2l+2m) +e?(6k?6l?2m) + e (8k+6l+2m) ; Z ppmm( k; l; m) = 4e?4k + 8e 4k + 2e 2k + 2e (4k?4l) + 2e?(4k?4l) +2e?(4l+4m) + 2e (4l?4m) + 4e?4m + 4e 4m +2e (8k?4l+4m) + 2e?(8k?4l?4m) : (2) Finally, use (2) in (), to obtain the recurrence relations, shown in (8) Phase diagram and free energy surface The reduced free energy y per spin for the tetrahedral gasket is a function of the partition function Z N : f s = (=N) ln Z N ; (3) y With F =?kbt ln Z =?= ln Z being the Helmholtz free energy, the reduced free energy is given as f =?(=N) F.

8 8 Exact two{dimensional renormalization group calculation where N is the number of spins in the system. The decimation 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 (7), we expand out (3), where N is the number of spins on the decimated lattice, each contributing k to the free energy. Note that k is a prefactor in (4), and Z the partition function, (5), 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 2. This, in essence, is the RSRG procedure, which yields a derivation of the free energy: f s = (=N) ln Z N = (=N) ln Z N = (=N) ln e [(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?) ) + ; (4) which is summarized as: f s = X i= (=4) i k i (k (i?) ; l; m (i?) ) ; (5) where k i is evaluated using (a), and the variables k i ; m i are the couplings; which for i, are obtained using (b) and (c), respectively (for instance, k in (5) 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 recurrence relations, (8), are used to track renormalization ows in the km{ plane. Basically, the ows track the renormalized couplings, k, and m, through as many steps as required. The ows can be generalized to produce a phase diagram of the models overall (renormalization) behavior. Flow calculations on the tetrahedral gasket were carried out for dierent values of second nearest{neighbor l-coupling. Specically, ows for values of l ranging between l =?: and l = : were investigated. Two separate regions in the phase space are easily identied, these regions being separated by a separatrix (shown as a dashed line on the phase diagram). Flows on one side of the separatrix go to a sink, while ows on the other side of the separatrix all go to a subspace (which is shown as a solid line on the phase diagram). Once the ows get to the subspace, the ows have diering behaviors based on the value of the l{coupling. When l >, the ows move out along the subspace towards increasing values of k. For l =, the ows remain stationary on the subspace for large enough initial values of the couplings. For the cases when l <, I nd that once the ows reach the subspace, they (i.e., the ows) progress along the subspace towards the critical point, eventually arriving at the sink. A plot of the phase diagram for the case of l = : is shown in Figure 3. The general features shown on the phase diagram include a sink at (k = :994; m =?:8), a critical point at (k = :525; m =?:28), the separatrix (shown as a dashed line), and the attractive subspace (shown as a solid line). The subspace

9 Two{dimensional fractal 9 m k Figure 3: Phase diagram for l = :. begins at the critical point and moves out in quadrant IV and eventually becomes linear at large k, where it is described by the equation: k =?m + : =?m + ln 2 : (6) 4 The critical point, (k ; m ), is found by looking for the location where the subspace and separatrix intersect. This point is found numerically and is given approximately as k ' :525 m '?:28 : (7) The phase diagram in Figure 3 is only valid for l = :. While the asymptotic limit for the attractive subspace is always given by (6), all other features of the phase diagram change when the value of the l-coupling is varied. For instance, for l = the sink moves to the origin, and for l =?: the sink locates at (k = :7435; m =?4:335?6 ). The separatrix also moves as l is varied, as seen by the fact that for couplings along the k-axis, a value of k :5 eventually ows out along the subspace when l = :, but for l = :, values of k :42 are required to obtain a ow which does not go to the sink. The reduced free energy per spin, (5), 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 4. 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. The trough in quadrants I and IV of the free energy surface corresponds to the separatrix of the phase diagram.

10 Exact two{dimensional renormalization group calculation m f(k,.,m) k 2 Figure 4: Free energy surface using l = : Critical exponents and Once the critical point and phase diagram have been established, linearizing about the critical point given in (7), followed by the identication of a relevant eigenvalue, enables the calculation of the thermal exponent y T. The relevant eigenvalue is associated with the eigenvector that points away from the separatrix. The Jacobian of the renormalization, (8), about the critical point in (7), is given (8) =@m k ;m :5457 :2968 = (9)?:47469?:567 The eigenvalues and eigenvectors of (9) are given as: = : = :6634 ; : V = V 2 = :9526?:346?:4997 :98869 : (2) The eigenvectors are shown on Figure 5 along with the separatrix. Notice that one of the vectors points towards the region containing the attractive subspace, while the other vector is tangent to the separatrix at the critical point. The relevant

11 Two{dimensional fractal m k Figure 5: Linearization around critical point - eigenvectors. vector is V. To obtain the thermal exponent, note that is related to the thermal exponent and dimensionality: b yt = ; (2) thus y T = (ln )=(ln 2) = :52864 ; (22) where b = 2 for our model. The critical exponents and are: = 2? d 2 = 2? y T :52864 =?:7833 = y T = = :896 ; (23) :52864 and the specic heat and correlation length follow using C t? and =a t?, respectively, where t (T c? T ) T c ; (24) and the relationship between k and T is given by scaling the coupling such that T = k : (25) 3.4. Hamiltonian for odd interactions In this section, (4) is expanded by including the odd interaction couplings, h and p. Therefore, the interactions for the base tetrahedron now include: k : a zero-spin coupling,

12 2 Exact two{dimensional renormalization group calculation 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 2 is derived in the same way as (3), and with odd interactions included is given by:?h = m (ss5s6s7 + s2s6s9s + s3s5s8s9 + s4s7s8s) + l (ss8 + ss9 + ss + s2s5 + s2s7 + s2s8 +s4s6 + s4s5 + s4s9 + s3s7 + s3s + s3s6) + 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) + h ((=2) (s + s2 + s3 + s4) + s5 + s6 + s7 + s8 + s9 + s) + p (ss5s6 + ss6s7 + ss5s7 + s5s6s7 + s6s2s9 +s6s2s + s6s9s + s2s9s + s5s9s3 + s5s9s8 +s5s3s8 + s3s9s8 + s7s8s + s7ss4 + s7s8s4 + s8ss4): (26) Note that the corner spins are given only (=2) of an h-coupling in (26). This counting scheme for both k and h is also evident in the Hamiltonian for the base tetrahedron:?h = 2 k + m (ss2s3s4) + k (ss2 + ss3 + ss4 + s2s3 + s2s4 + s3s4) + h (=2) (s + s2 + s3 + s4) + p (ss2s3 + ss2s4 + ss3s4 + s2s3s4) : (27) According the RSRG program, we desire the recurrence relations (cf. (8)): 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) : (28) Proceeding in a similar manner as when only even couplings were present, the equations (29) - (3) essentially give the solution to the recurrence relations, (28). Comparison of (29) with (9) shows that when odd interactions are introduced, ve independent partial partition functions are obtained, as compared to three when only even interactions are considered. Note specically that Z pppp 6= Z mmmm, and Z pppm 6= Z mmmp, when odd interactions are present.

13 Two{dimensional fractal 3 The ve independent equations for the renormalized couplings are ln[z pppp ] = 2k + 6k + m + 2h + 4p ln[z mmmm ] = 2k + 6k + m? 2h? 4p ln[z pppm ] = 2k? m + h? 2p ln[z ppmm ] = 2k? 2k + m ln[z mmmp ] = 2k? m? h + 2p : (29) Solving for the renormalized couplings gives the equations: k = (=32)(ln[Z mmmm ] + 4 ln[z mmmp ] + 6 ln[z ppmm ] +4 ln[z pppm ] + ln[z pppp ]) k = (=6)(ln[Z mmmm ]? 2 ln[z ppmm ] + ln[z pppp ]) m = (=6)(ln[Z mmmm ]? 4 ln[z mmmp ] + 6 ln[z ppmm ]?4 ln[z pppm ] + ln[z pppp ]) h = (=6)(? ln[z mmmm ]? 2 ln[z mmmp ] + 2 ln[z pppm ] + ln[z pppp ]) p = (=6)(? ln[z mmmm ] + 2 ln[z mmmp ]? 2 ln[z pppm ] + ln[z pppp ]) ; (3) where the partial partition functions, in terms of the original couplings, are Z pppp = 2e?4k?4l + 2e 4h+4k+4l + 3e?8k?4l+4m +3e 4h+4l?4m?8p + 2e 2h?4k?4p + 4e 2h?4m?4p +6e?2h?4k?8l+4p + 6e 6h+2k+8l+4p + 4e 2h+4m+4p +e?4h?2l?4m+8p + e 8h+24k+2l+4m+6p Z mmmm = 2e?4k?4l + 2e?4h+4k+4l + 3e?8k?4l+4m +e?8h+24k+2l+4m?6p + e 4h?2l?4m?8p + 6e?2h?4k?8l?4p +6e?6h+2k+8l?4p + 4e?2h+4m?4p + 2e?2h?4k+4p +4e?2h?4m+4p + 3e?4h+4l?4m+8p Z ppmm = 8e?4k + 4e?4h+4k + 4e 4h+4k + 4e?4k+4l + 2e?4l?4m +2e?8k+4l+4m + 2e 4k?4l?8p + e 4h+4l?4m?8p +e?4h+8k?4l+4m?8p + 3e 2h?4k?4p + e?6h+2k?4p +4e?2h+4k?4l?4p + 4e 2h?4k+4l?4p + 2e 2h?4m?4p +2e?2h+4m?4p + 3e?2h?4k+4p + e 6h+2k+4p +4e 2h+4k?4l+4p + 4e?2h?4k+4l+4p + 2e?2h?4m+4p +2e 2h+4m+4p + 2e 4k?4l+8p + e?4h+4l?4m+8p +e 4h+8k?4l+4m+8p

14 4 Exact two{dimensional renormalization group calculation Z pppm = e h+6k?6l?2m?p + 3e 3h?2k+2l?2m?6p + 3e 3h?2k+6l?2m?6p +3e?h+2k?6l+2m?6p + 6e h?2k?2l?2m?2p + 3e h?2k+2l?2m?2p +3e 5h+6k+6l?2m?2p + 3e?3h+2k?6l+2m?2p + 6e h?6k+2l+2m?2p +e h?6k+6l+2m?2p + e?5h+6k?6l?2m+2p + 6e?h?2k?2l?2m+2p +3e 3h+6k?2l?2m+2p + 3e?h?6k?2l+2m+2p + 3e?h?6k+2l+2m+2p +6e 3h+2k+2l+2m+2p + 3e?3h?2k?2l?2m+6p + 3e h+2k?2l+2m+6p +3e 5h+k+2l+2m+6p + e 7h+8k+6l+2m+p Z mmmp = e?7h+8k+6l+2m?p + 3e 3h?2k?2l?2m?6p + 3e?h+2k?2l+2m?6p +3e?5h+k+2l+2m?6p + e 5h+6k?6l?2m?2p +6e h?2k?2l?2m?2p + 3e?3h+6k?2l?2m?2p + 3e h?6k?2l+2m?2p +3e h?6k+2l+2m?2p + 6e?3h+2k+2l+2m?2p + 6e?h?2k?2l?2m+2p +3e?h?2k+2l?2m+2p + 3e?5h+6k+6l?2m+2p +3e 3h+2k?6l+2m+2p + 6e?h?6k+2l+2m+2p +e?h?6k+6l+2m+2p + 3e?3h?2k+2l?2m+6p +3e?3h?2k+6l?2m+6p + 3e h+2k?6l+2m+6p +e?h+6k?6l?2m+p : (3) 3.5. Critical behavior With the odd interactions present, the magnetic exponent, y H, can be obtained, eectively giving the complete set of critical exponents for the tetrahedral gasket. A direct numerical calculation of the magnetization (M =@h) near the critical point is also considered Linearization near critical point Near the critical point, we look for a relevant eigenvector that ows across the separatrix surface. With the associated relevant eigenvalue, the magnetic exponent y H is calculated. The value obtained for y H ; combined with the value of y T ; (22), allow the remaining critical exponents and to be found. For the full tetrahedral gasket model with odd interactions, the linearization looks like: k m h p =@p C A k ;m ;h ;p This technique is used in (8) for the system with even couplings to obtain the relevant eigenvalue from whence the thermal exponent was obtained. Linearizing (32) k m h p : (32)

15 Two{dimensional fractal 5 around the critical point, (7), in kmhp{space, the following matrix, eigenvalues and eigenvectors are obtained: :546 :297 : :?:4747?:57 : : :? 3:3844 3:569 :??:355 :297 C A ; (33) V = V 2 = V 3 = V 4 = = 3: = : = : = :6634 ; (34) +? :99893?:4634 :9526?:346 : :? +?:77378 :63346 :4997?:98869 : : Note that the eigenvectors V 2 and V 4, along with eigenvalues 2 and 4, reproduce our previous results, (2), for even interactions. The other eigenvectors, V and V 3, are shown in Figure 6. Consider also how these eigenvectors t into the free energy surface shown in Figure 7. The free energy surface on the hp{plane in Figure 7 is calculated with values of the couplings k, l and m, xed at values of k, : and m, respectively. The eigenvector V is the relevant eigenvector, lying along the direction of ows away from the critical point in the hp{plane. The magnetic exponent, y H, is calculated using from (34): b yh = ; C A C A C A C A : (35) thus y H = (ln )=(ln 2) = :68652 : (36)

16 6 Exact two{dimensional renormalization group calculation p h Figure 6: Linearization on the hp{plane - eigenvectors p 2 6 f(k*,m*,.,h,p) h 2 Figure 7: Free energy surface on the hp{plane.

17 Using (36) the critical exponents and are found to be: = 2? y H y T = :593 Two{dimensional fractal 7 = 2 y H? d y T = 2:5973 ; (37) and the magnetization and susceptibility then follow from M t and t?, respectively Numerical calculation of the magnetization The magnetization near the critical point can be calculated using a critical exponent, but is also related to the free energy through a rst derivative. Specically, for the tetrahedral gasket near the critical point: M t : (38) Using the free energy, f s (k; m; l; h; p) from (5), with k from (28), near the critical point, = lim h! f s (k; m ; l = :; h; )? f s (k; m ; l = :; ; ) h : (39) Now view M as a function of temperature, by using (25), and dening the critical temperature, T T c. Using (7) for the value of k, the critical temperature is T = k = = 6:5574 : (4) :525 A plot of the results for h = 6?7 is shown in Figure 8. The qualitative features of Figure 8 correspond to the expected result (for instance, see [2]). The abscissa has been normalized by dividing by the critical temperature, (4). The ordinate is the magnetization M. Evidently, is just the slope of the line of the magnetization in the ln M? ln t plane, i.e., = ln M ln t ' :567 ; (4) which agrees fairly well with the theoretical value shown in (37). 4. Discussion The tetrahedral gasket evidently is a full two-dimensional geometry with exact recurrence relations, even in the presence of a nite eld. The complete set of critical exponents (,,, ) for the tetrahedral gasket have been calculated for the case when l = :. The tetrahedral gasket has a single critical point, given by (7), which is the point where the values of the critical exponents are evaluated using the eigenvalue method. A separate numerical calculation of the critical exponent

18 8 Exact two{dimensional renormalization group calculation.8.6 M T/T* Figure 8: Magnetization of the tetrahedral gasket. gives a result in good agreement with the theoretical value given in (37a). The availability of a second even coupling, m, allows us to plot a free energy surface, which has features in common with the phase diagram. Specically, we are able to identify the separatrix on the free energy plot. Now note that this model is not as general as, say, a planar two-dimensional model, since only ferromagnetic order is possible for spins setup according to the tetrahedral gasket conguration. Consider the base tetrahedron (the corner spins of Figure 2), we easily see that if the system wanted to align antiferromagnetically, frustration would result after the rst two spins where aligned in opposite directions (e.g., if s = +, and s2 =?, we nd that s3 can not align opposite to both s and s2). We know from thermodynamics [] that at equilibrium the system minimizes the Helmholtz free energy, but the reduced free energy introduces a change of sign. Therefore, over the manifold of states in our canonical ensemble the system will maximize the reduced free energy. Relating this fact (maximizing the free energy), to the ows that move out along the attractive subspace towards a larger value of f s (but only for l > ), shows in eect that this subspace represents an equilibrium state for the tetrahedral gasket. Recall, from (6), that the even couplings are actually opposing each other, i.e., k '?m, on the attractive subspace. This leads to an obvious question: Question: Why isn't the subspace oriented in such a way that the couplings cooperate (i.e., k = +m), or at least have the same sign?

19 Two{dimensional fractal 9 Conjecture: The system maintains a certain amount of entropy when the couplings have values that place them on the attractive subspace. When the k coupling is extremely large, and intuition would expect all the spins to align together (i.e., become completely ordered), the fact that the renormalization ow stays on the subspace suggests that this is an ordered state with a non-zero entropy. The conjecture points out a desirable feature for a realistic model of a physical system (compare this to a model that does have a reachable zero entropy state). This setup, of opposing couplings (on the attractive subspace), suggests that if the tetrahedral gasket has some physical counterpart, then this system is of a physical nature that resists having all the \spins" of each local unit aligned. It should be noted that the exponents indirectly depend on l and directly on the critical point given by (7), which is the point about which the linearization (8) and (32) are taken. This critical point (for l = :) was obtained numerically by running ow calculations at small increments across the separatrix to nd a point which did not ow quickly away from the separatrix. One extension to this work would be to nd the critical point analytically (for a general value of l). A further extension is a complete analysis of how the dynamics of the system change as the value of the second-neighbor interaction coupling, l, is varied. This has been studied [8] for the Sierpinski gasket with some interesting results, which include an analysis of how the critical point varies as a function of l, eectively giving k (l). Acknowledgment I would like to acknowledge Dr. J.S. Walker at Washington State University for his patience and assistance oered during the completion of [], and also Dr. P.A. Deutchman at the University of Idaho, for his work as my virtual advisor. The careful reading and editing of earlier drafts by J. Ulinder has also been very helpful. References. J. F. Nystrom, An Exact Two-Dimensional Finite Field Real-Space Renormalization- Group Calculation, (MS Thesis, University of Idaho, 996). 2. B. Bonnier, Y. Leroyer, C. Meyers, Physical Review B, 37, 525 (988). 3. Y. Gefen, B.B. Mandelbrot, A. Aharony, Physical Review Letters, 45, 855 (98). 4. Y. Achiam, Physical Review B, 3, 4732, (985). 5. Y. Gefen, A. Aharony, B.B. Mandelbrot, Journal of Physics A, 7, 277, (984). 6. Y. Gefen, Y. Meir, B.B. Mandelbrot, A. Aharony, Physical Review Letters, 5, 45, (983). 7. Y. Wu, B. Hu, Physical Review A, 35, 44 (987). 8. J.S. Walker, C. Kwiatkowski, Exact phase transitions in a Sierpinski gasket with fractal periodic boundary conditions, (unpublished). 9. J.H. Barry, M. Khatun, T. Tanaka, Physical Review B, 37, 593 (988).. R.K. Pathria, Statistical Mechanics, (Pergamon Press, 972).. H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, (John Wiley & Sons, 985).

20 2 Exact two{dimensional renormalization group calculation 2. J.S. Walker, C.A. Vause, Journal of Chemical Physics, 5, 266 (983). 3. S.M. Wood, J.S. Walker, Journal of Physics A, 27, 37 (994). 4. M. Schick, J.S. Walker, M. Wortis, Physical Review B, 6, 225 (977). 5. R.B. Fuller, Synergetics, (Macmillan, 975). 6. J.C. Hart, Computer Display of Linear Fractal Surfaces, (PhD Thesis, University of Illinois at Chicago, 99). 7. B.B. Mandelbrot, The Fractal Geometry of Nature, (W.H. Freeman, 982). 8. J.T. Sandefur, The American Mathematical Monthly,, 7 (996). 9. G.M. Bell, D.A. Davis, Statistical Mechanics of Lattice Models Volume : closed form and exact theories of cooperative phenomena, (Ellis Horwood, 989). 2. S. Wolfram, Mathematica, (Addison-Wesley, 99). 2. N.W. Ashcroft, N.D. Mermin, Solid State Physics, (W.B. Saunders, 976).