# The absence of phase transition for the classical XY-model on Sierpiński pyramid with fractal dimension =2

###### Abstract

For the spin models with continuous symmetry on regular lattices
and finite range of interactions the lower critical dimension is =2.
In two dimensions the classical XY-model displays Berezinskii-Kosterlitz-Thouless
transition associated with unbinding of topological defects (vortices and antivortices).
We perform a Monte Carlo study of the classical XY-model on Sierpiński pyramids whose fractal
dimension is 4/2=2 and the average coordination number per site is 7. The specific
heat does not depend on the system size which indicates the absence of long range order. From the
dependence of the helicity modulus on the cluster size and on boundary conditions we draw a conclusion
that in the thermodynamic limit there is no Berezinskii-Kosterlitz-Thouless transition at any finite
temperature. This conclusion is also supported by our results for linear magnetic susceptibility.
The lack of finite temperature phase transition is presumably caused by the finite order of ramification
of Sierpiński pyramid.

Y-model; fractals; Sierpiński pyramid; Monte Carlo simulations

## 1 Introduction

One of the powerful predictions of the renormalization group theory of critical phenomena is universality according to which the critical behavior of a system is determined by: (1) symmetry group of the Hamiltonian, (2) spatial dimensionality and (3) whether or not the interactions are short-ranged [1]. The possibility of phase transitions on systems with nonintegral dimensionality was first considered by Dhar [2] for the classical XY-model and for the Fortuin-Kasteleyn cluster model on the truncated tetrahedron lattice with the effective dimensionality 23/5=1.365. No phase transition at any finite temperature was obtained. Subsequently in a series of papers Gefen et al. [3, 4, 5, 6] examined the critical properties of the Ising model (discrete symmetry) on fractal structures, which are scale invariant but not translationally invariant, in order to elucidate the relative importance of multiple topological factors affecting critical phenomena. They found that the Ising systems with given fractal dimension have transition temperature 0 if the minimum order of ramification , which is the minimum number of bonds that needs to be cut in order to isolate an arbitrarily large bounded cluster of sites, is finite. In the case of fractals with they presented arguments that 0 for the Ising model. Monceau and Hsiao [7] further studied the Ising model on fractals with using the Monte Carlo method and found weak universality in that the critical exponents depended also on topological features of fractal structures.

For the models with continuous symmetry, 2, on fractal structures Gefen et al. [5, 6] used a correspondence between the low-temperature properties of such models and pure resistor network connecting the sites of the fractal to argue that there is no long-range order at any finite temperature if the fractal dimension 2 even in the case of infinite order of ramification. Subsequently Vallat et al. [8] used the harmonic approximation to the XY-model ( symmetry) on two-dimensional Sierpiński gasket (=3, 3/2=1.585) to show that the energy of a vortex excitation is always finite and hence there is no Berezinskii-Kosterlitz-Thouless (BKT) transition [1] at any finite temperature as free vortices are always present. These conclusions were confirmed in a recent Monte Carlo study of the full XY-model on two-dimensional Sierpiński gasket [9].

Here we present a Monte Carlo study of the XY-model on three-dimensional Sierpiński pyramids with =4 and fractal dimension 4/2=2. The model is described by the Hamiltonian

(1) |

where 02 is the angle/phase variable on site , denotes the nearest neighbors and 0 is the coupling constant. In the case of translationally invariant system in two dimensions this Hamiltonian gives rise to BKT transition and we investigate if the same holds in the case of a fractal structure with fractal dimension 2.

The rest of the paper is organized as follows. In Section 2 we present our algorithm for generating Sierpiński pyramids and outline the Monte Carlo procedure for calculating the thermal averages. Section 3 contains our results and discussion and in Section 4 we give a summary.

## 2 Numerical procedure

The procedure which we used to generate three-dimensional Sierpiński pyramids (SP) is illustrated in Figure 1, which shows the transition from the zeroth-order SP (the tetrahedron of unit side) to the first order pyramid via translations by three nonorthogonal basis vectors =(1,0,0), =(0,1,0) and =(0,0,1). The pyramid of order +1 is obtained from the pyramid of order via translations by vectors 2, 2 and 2 (=0,1, ). It is clear that the number of vertices of the th order Sierpiński pyramid can be obtained from the recursion relation =4-6 with =4. Thus, in generating the pyramid of order +1 from the pyramid of order not every point of the th order pyramid gets translated by all three translation vectors 2, 2 and 2. The top of the th order pyramid, (0,0,0), is never translated. The remaining -1 points are then all translated by 2. Next, the same points except for (2,0,0) are translated by 2 and finally, all points except for (2,0,0) and (0,2,0) are translated by 2. For pyramids of order 9 we found it most efficient to represent a vertex by the number 10+10+10 and the three translation vectors 2, 2 and 2 by the numbers 210, 210 and 210, respectively. Then the result of translating a point represented by number P by a vector represented by is described by the number . From a given number representing a vertex it is easy to get its coordinates in the basis : /10, where denotes the integer part, 1010 and 1010.

In the Metropolis Monte Carlo scheme of calculating the statistical averages for a model with only the nearest neighbor interactions it is necessary to provide the list of nearest neighbors for each site/vertex. We take that all sites that are at a unit distance (the size of the edge of the elementary tetrahedron) from a given vertex are its nearest neighbors. Thus the number of nearest neighbors of site varies with the order of Sierpiński pyramid. For example, the vertex (1,1,0) has six nearest neighbors in the first order pyramid (Figure 1) but the same site has eight nearest neighbors in all higher order pyramids, the additional two neighbors being vertices (2,1,0) and (1,2,0). For the Sierpiński pyramids of orders 4,5,6 we found the average number of neighbors per site to be 6.923, 6.981, 6.995 with standard deviations 0.903, 0.876, 0.869, respectively. Thus, the average coordination number for three-dimensional Sierpiński pyramid is greater than the coordination number for three-dimensional simple cubic lattice. This fact should be kept in mind when we discuss our numerical results in the next section. In constructing the list of nearest neighbors for the sites in a pyramid we found it convenient to group all sites according to the values of the sum of their coordinates (i,j,k). The sites with the same belong to the same plane parallel to the basal plane of the pyramid defined by points (2,0,0), (0,2,0), (0,0,2), where is the order of the pyramid (see Figure 1). The nearest neighbors of a given site are located in the plane to which it belongs and in the neighboring planes. Throughout this work we employed two types of boundary conditions: closed, where the four corners of an th order pyramid were considered to be coupled to each other and open, where the four corners are uncoupled to each other.

The Monte Carlo (MC) simulation of the classical XY-model on Sierpiński pyramids was based on Metropolis algorithm [10]. We considered the pyramids of orders 4 (130 sites), 5 (514 sites) and 6 (2050 sites). For a pyramid of given order the simulation would start at a low temperature with all phases aligned. The first 120,000 steps per site (sps) were discarded, followed by seven MC links of 120,000 MC sps each. At each temperature the range over which each angle was allowed to vary [11] was adjusted to ensure an MC acceptance rate of about 50%. The errors were calculated by breaking up each link into six blocks of 20,000 sps, then calculating the average values for each of 42 blocks and finally taking the standard deviation of these 42 average values as an estimate of the error. The final configuration of the angles at a given temperature was used as a starting configuration for the next higher temperature.

## 3 Numerical Results and Discussion

The heat capacity per site shown in Figure 2 was calculated from the fluctuation theorem

(2) |

where is the Boltzmann constant, is the absolute temperature and denotes the MC average. The results did not depend on the type of boundary condition (closed or open) within the error bars in analogy to what was found for two-dimensional Sierpiński gasket [9]. In the same Figure we also show the results for heat capacity obtained for the XY-model on three cubic lattices with the periodic boundary conditions. The sizes of cubic lattices were chosen so that they are comparable to the sizes of three Sierpiński pyramids. The peak in the specific heat of cubic lattices near =2.2 increases with system size [12] as a consequence of diverging correlation length at a continuous (i.e. second order) phase transition. On the other hand the specific heat for the Sierpiński pyramids is virtually size-independent indicating the absence of long range order for the XY-model on three-dimensional Sierpiński pyramid at any finite temperature. Thus, although the average coordination number for the Sierpiński pyramid ( 7) is higher than its value for the cubic lattice, the topological properties of this fractal structure, in particular a finite order of ramification [5], are responsible for the lack of long range order at finite . The regular lattices have an infinite order of ramification as the number of bonds one needs to break in order to isolate an arbitrary large bounded cluster of lattice sites is infinite. For fractals with a finite order of ramification an arbitrarily large bounded cluster can be cut off from the rest of the structure by breaking off only a finite number of bonds and at finite temperature thermal fluctuations are sufficient to destroy the long range order.

The absence of size dependence of the peak in is found for the XY-model in two dimensions [13, 14]. In that case the peak results from unbinding of vortex clusters [13] with increasing temperature above the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature where the heat capacity has an unobservable essential singularity [15]. However the results in Figure 2 for fractal structures of fractal dimension 2 do not necessarily imply BKT transition since size-independent peaks in could result from the average energy per site changing monotonically from the values near -3.5 (each site has about 7 neighbors on average) at low temperatures to near zero in high temperature paramagnetic phase.

The main signature of BKT transition is a universal jump in helicity modulus =2/ [16] at critical temperature . The helicity modulus measures the stiffness of the angles with respect to a twist at the boundary of the system. At zero temperature, when the angles are all aligned, one expects a finite value of and at high temperatures, when the system is in disordered paramagnetic phase, vanishes. For the XY-model on three-dimensional regular lattices decreases continuously with increasing temperature and just below the transition temperature it obeys a power law [12]. In two dimensions, however, there is a discontinuity in stemming from unbinding of the vortex-antivortex pairs at BKT transition from quasi-long-range order (order parameter correlation function decays algebraically) to disordered phase (order parameter correlation function decays exponentially). In numerical simulations on finite systems discontinuity is replaced by continuous drop in which becomes steeper with increasing system size (see, for example, [9]). We should point out that since Sierpiński pyramid is three-dimensional object topological defects are not only vortices in planes parallel to the faces of the pyramid (which are two-dimensional Sierpiński gaskets) but also vortex strings. Kohring et al. [17] presented Monte Carlo evidence that a continuous phase transition for the XY-model in three dimensions is related to unbinding of vortex strings.

We computed the helicity modulus for Sierpiński pyramids following the procedure of Ebner and Stroud [18]. The idea is to think of Hamiltonian (1) as describing a set Josephson coupled superconducting grains in zero magnetic field, where is the phase of the superconducting order parameter on grain . Then if a uniform vector potential is applied the phase difference in (1) is shifted by , where is the position vector of site and is the flux quantum. The helicity modulus is obtained from the Helmholtz free energy as , i.e.

(3) |

We took to be along one of the edges of elementary tetrahedron (Figure 1). Our results for obtained with closed boundary condition are shown in Figure 3. They are completely analogous to what was obtained for two-dimensional Sierpiński gaskets [9]: a rapid downturn in starts near the universal 2/-line but the low temperature values of decrease with increasing system size and the onset of the downturn, which is in the vicinity of the putative phase transition, shifts to the lower temperatures. For the XY-model on the square lattices, where BKT transition does occur, the low- values of helicity modulus and the onset of its downturn do not depend on the system size (see, for example, [9]). Our results in Figure 3 suggest that in thermodynamic limit vanishes at any temperature 0 implying no BKT transition for the XY-model on Sierpiński pyramid of fractal dimension 2. This conclusion is reinforced by our results for obtained with the open boundary condition when four corners of an th order pyramid are not coupled to each other, Figure 4. The helicity modulus is zero within the error bars which are larger than those obtained with the closed boundary condition. The results in Figures 3 and 4 indicate that closed boundary condition introduces additional correlations as was the case for two-dimensional Sierpiński gaskets [9].

Our conclusion about the lack of finite temperature BKT transition is supported by results for linear susceptibility

(4) |

where is the magnetization of the system, shown in Figure 5. For finite cubic lattices one gets a peak in near 2.2 whose size and sharpness increase with the number of sites as a result of diverging correlation length at the onset of long range order. In the case of Sierpiński pyramids the peak in also grows with increasing system size but it also shifts substantially to lower temperatures. For BKT transition Kosterlitz predicted [19] that above the susceptibility diverges as , with =0.25, 1.5 and =0.5, and is infinite below . Our results suggest that in thermodynamic limit there would be no divergence in at any finite temperature for the classical XY-model on Sierpiński pyramid.

## 4 Summary

From our Monte Carlo simulation results we conclude that there is no finite temperature phase transition for the classical XY-model ( symmetry) on three-dimensional Sierpiński pyramid (fractal dimension =2). Since the heat capacity per site does not depend on the system size there can be no long range order at any finite temperature. Because the low-temperature helicity modulus decreases with increasing system size for closed boundary condition, and is zero within the error bars for open boundary condition, it must vanish in thermodynamic limit at any finite temperature. This implies no continuous finite temperature phase transition associated with unbinding of vortex strings [17] in which case the helicity modulus vanishes at transition temperature as a power law [12]. Moreover there is no finite temperature Berezinskii-Kosterlitz-Thouless transition associated with unbinding of vortices and characterized by discontinuity in at . These conclusions are supported by our results for linear magnetic susceptibility. The lack of finite-temperature long range order and the vanishing of spin stiffness/helicity modulus are the consequence of finite order of ramification of Sierpiński pyramid: as an arbitrarily large bounded cluster of sites can be disconnected by cutting only the finite number of bonds thermal fluctuations drive helicity modulus to zero and destroy long range order.

### 4.1 Acknowledgements

We thank Professor S. K. Bose for many useful discussions. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The work of M. P. was also supported in part through an NSERC Undergraduate Student Research Award (USRA).

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