# Phase diagram and exotic spin-spin correlations of anisotropic Ising model on the Sierpiński gasket

###### Abstract

The anisotropic antiferromagnetic Ising model on the fractal Sierpiński gasket is intensively studied, and a number of exotic properties are disclosed. The ground state phase diagram in the plane of magnetic field-interaction of the system is obtained. The thermodynamic properties of the three plateau phases are probed by exploring the temperature-dependence of magnetization, specific heat, susceptibility and spin-spin correlations. No phase transitions are observed in this model. In the absence of a magnetic field, the unusual temperature dependence of the spin correlation length is obtained with JJ, and an interesting crossover behavior between different phases at JJ is unveiled, whose dynamics can be described by the JJ-dependence of the specific heat, susceptibility and spin correlation functions. The exotic spin-spin correlation patterns that share the same special rotational symmetry as that of the Sierpiński gasket are obtained in both the plateau disordered phase and the plateau partially ordered ferrimagnetic phase. Moreover, a quantum scheme is formulated to study the thermodynamics of the fractal Sierpiński gasket with Heisenberg interactions. We find that the unusual temperature dependence of the correlation length remains intact in a small quantum fluctuation.

###### pacs:

05.70.Fh and 75.10.Hk and 05.65.+b and 64.60.al## 1 Introduction

The classical models like Ising or Potts Potts () models often exhibit intriguing properties, especially when there exists geometrical frustration Frustrate () that may cause extensive ground-state degeneracy and nonzero residual entropy of the system at zero temperature ReEntro (). There are many interesting phenomena in these models that have been discovered but not yet been totally understood, for instance, the essence of the partial order PO () or the crossover between exotic phases, etc. Extensive works have been done on the Ising models in two- and three dimensions, such as the Ising models on the checkerboard lattice PO () and kagomé lattice liw (), as well as the Potts model on some irregular lattices POIrregular (); zhaoy (), where the systems possess translational invariance. There were some early works on the Ising model on fractal lattices Fractal1 (); Fractal11 (); Fractal12 (); Fractal2 (); Fractal3 (); Fractal4 (); Fractal5 (); Fractal6 (), which show the properties quite different from the models on one-, two- and three-dimensional lattices with translational invariance even with the simplest couplings. Recent reported on a new class of highly frustrated two-dimensional magnetic materials (cpa=2-carboxypentonic acid, a derivative of ascorbic acid; X=F, Cl, Br) Material1 (); Material2 (); Material3 (); Material4 (); Material5 (); Material6 () stimulate study of the spin-Heisenberg model on triangulated kagome lattice(TKL) Lattice1 (); Lattice2 (); Lattice3 (); Lattice4 (); Lattice5 (); Lattice6 (); Lattice7 (). The TKL is actually an example of two-dimensional triangles-in-triangles (TIT) lattices TIT1 (); TIT2 (); TIT3 (), which generally consist of smaller triangular entities embedded in either some or all triangular cells of 2D triangle-based lattices. Latest study on spin- Ising-Heisenberg model on two closely related TIT lattices reveal a profound relation between local quantum fluctions with overall magnetic behavior of this mixture classical-quantum spin model TIT1 () and shed light the role of classical-quantum spin model on research the ground state of quantum Heisenberg model TIT2 (). But for the Ising models on fractals, comprehensive studies on, e.g. spin-spin correlations where there is no translational invariance, are still needed and, the questions, such as how to describe the crossover behaviors in fractals, are yet to be explored.

The Sierpiński Ising model is so defined that the Ising spin is settled on each of vertices of the Sierpiński graph. The Hausdorff dimension of an Sierpiński graph satisfies , which is not an integer, where is its Euclidean dimension. Unlike other lattices such as square or kagomé lattices, a Sierpiński graph has no translational symmetries, but bears some special geometrical properties that can be seen from its recursive construction procedure. The Sierpiński graph, which is also named Sierpiński gasket (SG), is shown schematically in Fig. 1 (a).

Some properties of the Ising model on Sierpiński graphs were noted. For example, the order of ramification on a SG is “marginal” Fractal12 (), and by locating the fixed point of renormalization group (RG) equations, it was confirmed that there is no phase transition at any finite temperature Tc (). With the nearest neighbor ferromagnetic couplings, the correlation length increases extremely fast with the inverse temperature as Fractal12 ()

(1) |

where is a positive constant (ferromagnetic coupling) and Boltzmann constant is taken as . This unusual relation indicates that the correlation length is finite at any finite temperature. Since increases so fast with , upon lowering temperature, the correlation length can be larger than the size of any finite system, resulting in a long range order.

In this paper, the anisotropic antiferromagnetic Ising model on the SG is studied systematically in the thermodynamic limit. The ground state phase diagram in the plane of magnetic field ()-interaction (J) is obtained, where three nontrivial phases, including the magnetization plateau ordered and disordered phases, and the plateau partially ordered ferrimagnetic phase, as well as five boundaries and two intersection points are identified. The plateau disordered phase and the plateau partially ordered ferrimagnetic phase are found to have extensive degeneracy. The thermodynamic properties of the system, including the temperature dependence of the uniform and subset magnetization, susceptibility, specific heat and spin correlation functions, are calculated. Some interesting results due to special geometrical properties like the fractional dimensionality and the absence of symmetries on usual periodic lattices are discovered. The nonexistence of the phase transition is testified. For , the unusual relation between the correlation length and temperature is obtained for JJ, and a nontrivial crossover behavior between different phases at JJ is disclosed, whose dynamics can be described by the JJ-dependence of the specific heat, susceptibility and the spin correlation functions. Exotic spin-spin correlation patterns that have the same special rotational symmetry as that of the SG are observed in the plateau disordered phase. Moreover, we propose a scheme to study the thermodynamics of Sierpiński gasket with quantum interactions, and discover that the unusual relation of correlation length remains intact in a small quantum fluctuation.

This paper is organized as follows. In Sec. II, we introduce the anisotropic SG Ising model and give the exact formulation of the free energy. In Sec. III, the zero temperature phase diagram is identified by calculating the ground state uniform magnetization and residual entropy Residual (). In Sec. IV, the temperature dependence of thermodynamic quantities, including the uniform and subset magnetization, susceptibility and specific heat, are studied. In Sec. V, the exotic spin-spin correlation functions of the three nontrivial plateau phases are disclosed. In Sec. VI, we generalize the scheme to study the thermodynamics of quantum models on the fractal Sierpiński gasket. Finally, a summary is given.

## 2 Free energy of anisotropic Ising model on Sierpiński gasket

Let us begin with discussing the exact formulation of free energy of the anisotropic antiferromagnetic Ising model on the SG with . The Hamiltonian reads

(2) |

where means the spins at vertices and that are nearest neighbors on the SG, each spin takes values of , is the magnetic field and is the negative coupling constant. The couplings are indicated in Fig. 1 (a), where we presume for the blue bonds and for the red bonds. We denote the SG that contains spins as and its density matrix as , where is the renormalization step. Then, the partition function can be expressed as .

We begin with , which is the density matrix of the model with only six Ising spins in [Fig. 1 (b)] as

(3) |

The density matrix for can be represented by

(4) |

Now, we introduce the “corner density matrix” by summing over the degrees of freedom of except for the spins at the three vertices of the largest triangle in the . For example, we have , where , and are the three spins at the three vertices of the largest triangle in the . The recursive relation of the real space renormalization of the corner density matrix can be given by decimating the degrees of freedom inside [Fig. 1 (b)] as

(5) |

where , and denote the spins at the corner of the largest triangle in . As a matter of fact, Eq. (5) can be considered as tensor contractions. This real space renormalization method is equivalent to a tensor renormalization group approach TRG () on a fractal lattice. The factor is introduced to avoid divergence.

The partition function of the Ising model on with spins becomes

(6) |

with . So, the free energy per site can be obtained by

(7) |

When , one can take , and because is finite.

The thermodynamic quantities such as the energy per site, specific heat, entropy, uniform magnetization and susceptibility can be calculated through the differentiation of the free energy given in Eq.(7).

Moreover, the magnetization of the single spin and the spin-spin correlation function , defined by

(8) | |||||

(9) |

can be obtained by performing the calculations similar to Eq. (5). We take as an example. Considering that the only difference between the equations for and is the contraction that involves , we can introduce the impurity as , and then, its renormalization can be written as

(10) |

where each satisfies Eq. (5). Compared with Eq. (5), one of the three ’s in Eq. (10) is replaced by the impurity, and which tensor should be replaced depends on the position of in the SG. In the end we have . The calculation of is similar. It should be mentioned that the whole formulation above can be readily extended to the N-state Potts model Potts (); zhaoy ().

The above formulation can be further simplified in the absence of a magnetic field (), and the fixed point of the renormalization can be discussed. Under this circumstance, each tensor only contains two inequivalent components denoted as and . According to Eq. (5), we can readily get the recursive relation for as Recursive ()

(11) |

with . Consequently, the free energy becomes

(12) |

Eq. (11) has two fixed points: (unstable) and (stable). By considering that represents the probability distribution of an effective Ising model of the three spins, one can see that implies the effective temperature (infinite inverse temperature), and implies when all configurations of spins share the same probability, indicating that no phase transition happens at any finite temperature. This is consistent with the previous result in e.g. Tc ().

## 3 Phase diagram at zero temperature

By calculating the residual entropy and the uniform magnetization in a magnetic field at sufficiently large inverse temperature RGT (), we have found four phases that are marked by A, A, A and A (Fig. 2). For and (we take as energy scale and for the antiferromagnetic coupling), the system is in the A phase, where the residual entropy and the uniform magnetization . After grouping the spins into four subsets denoted as P, P, P and P [Fig. 4 (a)], we calculated the zero temperature subset magnetization and obtained and with () the subset magnetization of . It can be seen that the system is in the ferrimagnetic order, where of the spins that point up and belong to P and P subsets, and of them that point down and belong to P and P subsets, resulting in a magnetization plateau phase.

Keeping and increasing the coupling from to , we observed that the system enters into the A phase (e.g. , ) with the same magnetization , but the residual entropy is nonzero, , showing that this phase is in a disordered state with extensive ground state degeneracy. The spin configurations of elementary cell of the macroscopically degenerate -plateau state shows in Fig. 3. The subset magnetization in A phase is and , which is totally different from those in A phase. It can be seen that the residual entropy in A phase is associated with all spins, so A phase is of a disordered plateau phase. This fact indicates that by increasing , the ferrimagnetic order in A phase is “melted” in A phase by the classical frustration whereas the uniform magnetization is unchanged.

Noting that there are also exist plateau in both the spin- Ising-Heisenberg model and the pure quantum spin- Heisienberg model on two closely related TIT lattices at zero temperature in Ref.TIT2 ().We true believe that the ¡°triangles-in triangles¡± structure is responsible for the presence of plateau. This is illuminating to explore the effects of geometric frustration. For and , the system is in the A phase with and , and the subset magnetization satisfies and . The spins belonging to P and P are totally polarized, and only the spins belonging to P and P make contributions to the residual entropy. Thus, this phase is a partially ordered plateau ferrimagnetic phase.

For and , the system is in the A phase with and , which is nothing but a polarized state.

For , when the system bears the ferrimagnetic order with spontaneous magnetization , but on the contrary, when [the boundary in Fig. 2 (c)], we found the subset magnetization becomes zero, which indicates that may be a crossover line. Such a crossover behavior will be re-examined by studying the specific heat, susceptibility and spin-spin correlation functions in Secs. IV and V. In addition, it appears that the magnetization in A phase cannot appear spontaneously but can only be induced by a magnetic field.

It should be point out that in our calculation, we find that the boundaries of phases are continuous and there is no singularity at any finite temperatures. However, when we reduce the temperature, we find that the magnetization approaches to a step function,which implies a first-order transition driven either by the external magnetic field or the interaction term .

We also calculated the residual entropy at each of phase boundaries [Fig. 2 (c)], and disclosed that are higher than those of the surrounding phases. At boundary with and which is the left boundary of A phase, we got ; at boundary between A and A phases with and , we found ; at boundary between A and A phases with and , we had ; at boundary between A and A phases with and , we obtained ; at boundary between A and A phases with and , we arrived at . In particular, we observed that at the two intersection points of the boundaries are larger than those at the related boundaries, namely at the intersection point () between and , and at the intersection point () between , , and .

## 4 Thermodynamic Properties

In this section, we shall study the temperature dependence of the thermodynamic quantities including the uniform and subset magnetization, specific heat, susceptibility, where the nonexistence of phase transitions is testified, and the thermodynamic properties of the system will be explored.

Let us first look at the uniform magnetization [Fig. 4 (b)] and subset magnetization [the inset of Fig. 4 (b)] versus in A phase by taking and . One may see that a magnetization plateau appears at low temperature, and the magnetization becomes zero smoothly from nonzero at around , which indicates that the system is crossing from the low-temperature ferrimagnetic phase to the high temperature disordered paramagnetic phase.

The uniform and subset magnetization in A and A phases [Fig. 5] also have the similar behaviors, showing that the system is crossing into the high-temperature phases. In A phase, the spins belonging to P and P exhibit similar behaviors, where the subset magnetization first rises from zero to its maximum at about and then decreases to zero with increasing , while the spins at P and P sites behave similarly, where the saturation magnetization decreases to zero with increasing . In A phase, the situation is somehow different, where the spins at P and P sites are gradually decreases from polarized to zero , and the subset magnetization of P and P first converges to and then reaches the maximum as increases.

Fig. 6 shows the specific heat and the susceptibility versus in three nontrivial A, A and A phases. It indicates that the system enters the high-temperature phases through crossover instead of phase transitions (even though the peak in A phase is relatively shaper, it is still round and continuous). One may notice that the temperature at which the peak of susceptibility appears is slightly higher than that of the peak of specific heat, where the fluctuation of energy reaches the maximum.

In Sec. III, we have identified as a crossover line for . Here, we studied this crossover behavior by calculating the specific heat and susceptibility for different , as shown in Fig. 7. For (including the point ), there is only one peak of at high temperature which is independent of . With gradually decreasing to , another peak moves from the low-temperature side to the high-temperature side, while the position of the high-temperature peak remains unchanged. This double-peak structure indicates a mixture of two kinds of excitations corresponding to the two phases. For smaller than , the two peaks merge into one, and the low-temperature peak stands dominant. For the susceptibility , there is only one peak, whose position moves to infinitesimal as . This peak, although it looks apparently divergent, is still finite and broad. Meanwhile, as shown in Fig. 7 (b), the relation between and becomes more and more linear as . When , bears a linear relation with . In the following section, we will show for different , the positions of both the low-temperature peaks of the specific heat and the susceptibility locate within the finite correlated region.

## 5 Exotic spin-spin correlations

Now we study the spin-spin correlation function [Eq. (9)] in the three non-trivial phases. The A phase is manifested to be with a Ferrimagnetic order, and at it has an unusual long range correlation (ULRC) which is similar to the behavior characterized by Eq. (9) and the long range correlation is in the usual sense in the presence of . We also discovered that the ULRC corresponds to an intermediate region which separates the diagram into two regions, the fully correlated region with the correlation length and the non-correlated region with . In the ULRC region where the system is finitely correlated, increases “super-exponentially” with until the system enters into the fully correlated region. When , and , the upper and lower boundaries of the ULRC region, both approach infinite, and so consequently, the system can only stay in the non-correlated region at finite temperature for .

The correlations in the A and A phases are more complicated, where exotic correlation patterns that have same special rotation symmetry as that of the SG are disclosed below. The A phase is a fully polarized phase and will not be discussed here.

### 5.1 Unusual long range correlation for and

The spin-spin correlation functions versus for and for different distances between two spins are studied. Fig. 8 (a) shows the results at . By defining the distance where the system bears the correlation as the correlation length , we determine the -dependence of . We discovered that in this ferrimagnetic phase for smaller than a certain value denoted as , the correlation length shares the similar unusual form as Eq. (9), where increases extremely fast when increases. We show the high linearity of the relation between and , i.e. [Fig. 8 (b)], which gives rise to the temperature dependence of the correlation length having the simple form of

(13) |

where the coefficients and are determined by and . The values of and for several ’s are shown in Table LABEL:tab-linear for useful information. Also, it should be noted that the regression coefficient of each linear fitting is larger than . The correlation function is independent of the distance in the presence of .

0 | 0.2 | 0.5 | 0.9 | |
---|---|---|---|---|

0.8547 | 0.6907 | 0.4206 | 0.0835 | |

-0.375 | -0.2462 | -0.054 | -0.0251 | |

0.9995 | 0.9994 | 0.9996 | 0.9995 |

Also, by locating the inverse temperature where the correlation length is , we can find where the system begins to be correlated. Fig. 9 shows the -dependence of and , where the whole - region is separated into three subregions. In the subregion beyond the line, the system is fully correlated with ; in the subregion below the line, the system is non-correlated with ; in the subregion between two lines where the system is finitely correlated, the correlation length obeys Eq. (13) in which increases “super-exponentially” with .

From Fig. 9, the behavior of and when as well as can be obtained. With , the two curves manifest a linear relation with . With from the side, both and go divergent with the speed faster than the exponential divergence. Such a behavior of can provide a clear picture of what happens during the crossover at in the absence of a magnetic field. With , the system needs a much lower temperature crossover to the finitely correlated region. At , as becomes infinite, the system is forbidden to crossover to the finitely correlated region at any finite temperature. Thus, is a crossover line. Meanwhile, this is coincident with the results of the low-temperature peaks of and shown in Fig. 7, which stand within the finitely correlated region.

For and [the boundary, Fig. 2 (c)], the correlation functions with is zero even at sufficiently large , which is coincident with previous results.

### 5.2 Exotic correlation patterns in A and A phases

In this subsection, we discuss the spin-spin correlation functions in other two nontrivial A and A phases with sufficiently large .

By introducing with and (), we categorize the spin-spin correlations in A phase into ten cases (, , , , , , , , , ) and discuss them separately in the following.

(1) for any choice of two spins that belong to the subset P.

(2) when and in the same , or otherwise. See Fig. 10 (a).

(3) when and in the same , when and in two different but adjacent ’s, or otherwise.

(4) when and in the same or otherwise.

(5) It is shown in Fig. 10 (b) that is equal to either or when are in the same , and is equal to either or when and are in two different but adjacent ’s. for any choice of and that are in two different ’s which are not adjacent. Thus for the long distance, we have the correlation which is induced by the magnetic field.

(6) The pattern of within one or two adjacent ’s is similar to that of as shown in Fig. 10 (c). For the long distance, we have when and are in two non-adjacent ’s, which is the same as .

(7) As shown in Fig. 10 (d), is equal to either or when , are in the same or in two different but adjacent . when and are in two non-adjacent ’s.

(8) when are in the same , and or when and are in two adjacent ’s, as shown in Fig. 10 (e), and when and are in two non-adjacent ’s.

(9) The pattern of within one or two adjacent ’s is similar to that of as shown in Fig. 10 (f). For a longer distance, , which is the same as .

(10) only when are in two different but adjacent ’s and also in two different but adjacent ’s, or otherwise (if and are in two different ’s but these two ’s belong to the same , ).

In A phase, the spins belonging to the subset P and P are polarized, so any two spins and with are maximally correlated with . We also calculated the correlation associated with the spins in P and P. The results show that with and we have , and with we have when , are the nearest neighbors, and when these two spins are not the nearest neighbors.

Despite the complexity, the correlation patterns share a common feature: for any , the pattern of the spin-spin correlations of and that belong to the bears the symmetry as same as in the SG. Specifically speaking, both the SG and the corresponding correlation pattern are invariant under the rotations with the angles around the center of as the rotating axis.

## 6 Generalization to quantum system

Below, we provide a TN-based scheme to study the finite temperature properties of the fractal Sierpínski gasket with quantum interactions and discuss if the super-exponential relation is still retained with the presence of a small quantum fluctuation. The quantum Hamiltonian reads

(14) |

where the first term is the Ising Hamiltonian along the z-direction of the spins and the second term gives the nearest-neighbor interactions that introduce quantum fluctuation. The finite temperature density operator is written as

(15) |

with the inverse temperature and the local quantum Hamiltonian of the th triangle. Then we employ the Trotter-Suzuki decomposition Trotter () as

(16) |

with a small quantity satisfying and containing higher-order small terms that originate from the commutation. The density matrix can be transformed into a TN form as followed. Introduce the local imaginary time evolution operator

(17) |

where is the coefficient matrix and () denotes the bra (ket) basis of a spin. The coefficient matrix is identically transformed into the product of a third-order tensor and three identities [Fig. 11 (a)] as

(18) |

Then is represented by a fractal TN form dubbed as fractal tensor product density operator (fTPDO) ODTNS () which reads

(19) |

with and meaning to trace all shared indexes. With the TN representation of , the finite temperature density matrix can be achieved by the imaginary time evolution algorithm iTEBD ().

Now we consider a simpler case with , implying that the quantum fluctuation terms can be treated as a perturbation. Then for any can be straightforwardly written in the fTPDO form by following exactly the same scheme that is given above. After summing over physical indexes by [Fig. 11 (b)], the partition function becomes the contraction of a fractal TN as

(20) |

which can be handled in the same way as the Ising case.

It is interesting to see whether the super-exponential relation that is observed in the Ising case will be destroyed by a small quantum fluctuation. One can observe from Fig. 12 that for different and , the relation of Eq.(13) still holds, though the correlation length is decreased by the quantum fluctuation especially for .

## 7 Summary

In summary, we have systematically studied the anisotropic antiferromagnetic Ising model on the fractal Sierpiński gasket where exotic properties due to both the classical frustration and the special geometry of the Sierpiński gasket are disclosed. The zero temperature phase diagram with four phases is presented, and the thermodynamic properties of the three nontrivial phases are explored by calculating the magnetization, residual entropy, specific heat and magnetic susceptibility. The spin-spin correlations are also probed. For , the thermodynamic crossover behavior with and the zero-temperature crossover from to are both analyzed by studying the temperature dependence of the correlation length. The exotic correlation patterns are disclosed in the magnetization plateau disordered phase and the plateau partially ordered ferrimagnetic phase. Furthermore, we generalized our scheme to study the fractal Sierpiński gasket with Heisenberg interactions, with which we discover that the super-exponential relation of the correlation length holds after introducing a quantum fluctuation.

## Author contribution statement

Shi-Ju Ran and Meng Wang made a major contribution to this paper. All other authors designed the research and write the manuscript equally.

The authors are indebted to W. Li, Z. C. Wang, X. Yan, Z. G. Zhu and C. J. Chen for useful discussions. This work is supported in part by the NSFC (Grants No. 90922033 and No. 10934008), the MOST of China (Grant No. 2012CB932900 and No. 2013CB93 3401), and the CAS.

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