Thermodynamics of Ising Spins on the Star Lattice
There is a new class of two-dimensional magnetic materials polymeric iron (III) acetate fabricated recently in which Fe ions form a star lattice. We study the thermodynamics of Ising spins on the star lattice with exact analytic method and Monte Carlo simulations. Mapping the star lattice to the honeycomb lattice, we obtain the partition function for the system with asymmetric interactions. The free energy, internal energy, specific heat, entropy and susceptibility are presented, which can be used to determine the sign of the interactions in the real materials. Moreover, we find the rich phase diagrams of the system as a function of interactions, temperature and external magnetic field. For frustrated interactions without external field, the ground state is disordered (spin liquid) with residual entropy per unit cell. When a weak field is applied, the system enters a ferrimagnetic phase with residual entropy per unit cell.
pacs:75.10.Hk, 75.30.Kz, 75.40.-s, 64.60.-i
Spin systems with geometrical frustration have both fundamental and practical importance. Theoretically, lots of interesting phenomena have been found in the geometrically frustrated systems, like the antiferromagnetic triangular lattice, kagome lattice. The systems can remain disordered even at absolute zero temperature because of the competitive magnetic interactions. For example, an antiferromagnetic triangular lattice has a residual entropy per unit cell Wannier (1950). The frustration effect has important application in achieving a lower temperatures through the adiabatic demagnetization compared with other methods. When the temperature, external magnetic field, and other factors are considered, the geometrically frustrated systems can show very rich phase diagrams. Diep (2004) A typical case is that the magnetocaloric effect can be enhanced near the phase transition points when a finite external field is applied. Zhitomirsky and Tsunetsugu (2004); Isakov et al. (2004); Aoki et al. (2004) Of considerable interest has been searching for geometrically frustrated systems. Some new frustrated materials have been fabricated and studied recently, such as , , and . Rosenkranz et al. (2000); Harris et al. (1997); Maruti and Ter Haar (1994); Mekata et al. (1998); Loh et al. (2008, 2008); Yao et al. (2008)
In 2007, a new class of geometrically frustrated magnetic materials polymeric iron (III) acetate Zheng et al. (2007) was fabricated, in which Fe ions form a two-dimensional lattice referred as star lattice. Experiment has found that the materials exhibit spin frustration and have two kinds of magnetic interactions: intratrimer and intertrimer shown in Fig. 1. The Fe ion has a large spin, which is . The system may have the paramagnetic ground state because of the geometrical and quantum fluctuations.
The Ising model on the star lattice with uniform ferromagnetic couplings has been solved. Barry and Khatun (1995) The critical temperature was given with . The Bose-Hubbard model on the star lattice has also been studied using the quantum Monte Carlo method. Isakov et al. (2009) Recently, the edge states and topological orders were found in the spin liquid phases of star lattice. Huang et al. (2012) Even though the quantum Heisenberg model is considered to be appropriate to help studying the new material because of its quantum fluctuations in the ground state, Richter et al. (2004) the Ising model on the star lattice is still very important especially for the non-uniform case. In real materials, the Fe ion has which is close to the classical limit and the magnetic system shows two types of interactions. Zheng et al. (2007) Therefore, it is important to study Ising spins on the star lattice with the asymmetric interactions, especially for the frustrated case.
In this paper, we aim at the thermodynamics of Ising spins on the star lattice with asymmetric interactions using the exact analytic methods and Monte Carlo simulations. We present the phase diagram as a function of interactions, temperature and external magnetic field. There is a clear difference between the case and the case. Our study provides useful information for determining the sign of .
This paper is organized as follows. The model is described in Sec. II. In Sec. III, we map the star lattice to honeycomb lattice and get the exact results. Sec. IV presents the phase diagrams as functions of interactions and external magnetic field. In Sec. V, the Monte Carlo results for the heat capacity and susceptibility are given. Finally, in Sec. VI we summarize the results.
The structure of star lattice and its unit cell are illustrated in Fig. 1. We study Ising spins on the star lattice with two kinds of nearest-neighbor interactions, the intratrimer coupling and intertrimer coupling . The Hamiltonian is
where runs over all the nearest neighbor spin pairs, is the intra-triangular interaction and is the inter-triangular interaction, is the external magnetic field, and . The unit cell of the star lattice contains six spins shown in Fig. 1. If we use and to denote the total numbers of - and -bonds, we have . The analytic result of partition function is obtained for .
For simplicity, we use as the units of energy in the following. The corresponding phase diagrams are actually in a three-dimensional parameter space, , and .
Iii Exact solution in zero field
In this section, we study the exact analytic results of Ising model on the star lattice in zero magnetic field (). Using a sequence of transformation and series reductionsLoh et al. (2008), we can transform the Ising model on the star lattice into one on the honeycomb lattice whose partition function has been exactly solved using the Pfaffian method Fisher (1966); Kasteleyn (1963). Besides the exact analytical results, we expand the partition function in series for some special cases.
iii.1 Effective coupling on the equivalent honeycomb lattice
iii.2 Phase boundary
It is known that the critical temperature of the honeycomb lattice Ising model is given by . Baxter (1989) Having mapped star lattice to honeycomb lattice, we can substitute this into the equivalent coupling in Eq. (6). Thus, an implicit equation for the critical temperature of star lattice Ising model can be obtained as,
This result is plotted in Fig. 3. When is ferromagnetic () and strong enough, the critical temperature saturates at a finite value, i.e. . The critical temperature drops to zero as approaches zero. When , the curve is approximately linear with .
Since all the factors obtained here are analytical, the singularity in the partition function remains when we transform the star lattice into the honeycomb lattice. The phase transition is the same as the honeycomb lattice where a continuous second-order transition happens.
If is antiferromagnetic (), we get the negative critical temperature, which implies no phase transition existing in this case. It can be used as a criterion for experimentalists to determine if the couplings in a real material is ferromagnetic or antiferromagnetic. If one finds a phase transition in the real material, we can conclude that the couplings and should be both ferromagnetic. In there is no long range order found, it means that at least one kind of nearest neighbor couplings is antiferromagnetic in the material.
iii.3 Partition function
Since we have utilized the transformation and series reductions to map the star lattice to a honeycomb lattice, the partition function per unit cell, , of the star lattice Ising model is equivalent to that of the honeycomb lattice multiplied by some coefficients which result from the transformation. These coefficients are as follows,
Therefore, the total partition function of the star lattice is
where is calculated using the Pfaffian method.Kasteleyn (1963) We rewrite it here,
We can rewrite and get a more accurate numerical evaluation according to the singularities of the integrand.
The partition function of the star lattice Ising model is therefore
The total partition function is given by , where is the spin number of unit cell. Since the partition function is obtained, the internal energy, specific heat, entropy and free energy can be calculated from it.
Taking derivation of the partition function, the energy per unit cell of the star lattice Ising model can be obtained.
where is expressed in terms of the complete elliptic integral of the first kind, K, Horiguchi et al. (1992)
iii.5 Specific heat
In the unfrustrated case, the specific heat shows a sharp peak at where a phase transition happens. The phase transition point is consistent with the result of Eq. (6). In addition, there is a broad hump at higher temperature because of the flopping of spins. Moreover, this hump changes with . It is obvious when and becomes indistinct when . However, it arises again when In the unfrustrated case, the sharp peak vanishes which implies no phase transition, consistent with the conclusion drawn from the phase diagram.
iii.6 Zero-temperature limit: residual entropy
In the case of , the partition function can be expanded as
The residual entropy is therefore when .
However, when , the model becomes frustrated. In this way, when , , which means , . Therefore, becomes . Expanding , we get
These results contribute to the residual entropy by
Thus, the frustration of the system leads to a residual entropy per unit cell when . One can confirm that, this value is consistent with the entropy at in Fig. 5. The residual entropy per site is approximately , smaller than the triangle lattice, TKL, and kagome latticeWannier (1950); Loh et al. (2008); Kanô and Naya (1953). Therefore, the star lattice is less frustrated compared to them.
Iv Phase diagrams at zero temperature
In this section, we present the phase diagrams at zero temperature along with some thermodynamic properties such as energy, magnetization and entropy. By calculating the ground state energy of the star lattice, we derive the full phase diagram for the system. Since the phase diagram at zero field is already shown in Fig. 3, we focus on the none-zero field case in this section. The corresponding results are summarized in Figs. 10 and 11 according to the sign of .
iv.1 Zero Field (Phase V and VI)
The phase diagram for zero field as a function of couplings is showed in Fig. 3. When , the phase is ordered and ferromagnetic. When , the phase is frustrated with a residual entropy . When , the phase is located in the negative section of -axis , which reveals that there is no phase transition in this phase. The disordered and ordered phases are labeled by V and VI in Figs. 10 and 11 respectively.
In phase V, the system is fully frustrated. We find degenerate ground states for each unit cell. However, as shown in Eq. (24), the residual entropy is not but . This is similar as the triangular lattice whose residual entropy can not be obtained by counting the number of ground states in a unit cell. Wannier, 1950
iv.2 Saturated ferromagnetic phase (Phase I)
When the external field is strong enough, i.e. , the phase is ferromagnetic where all spins are lined up. It is obvious that this state has energy , magnetization and entropy per unit cell.
iv.3 Phase II
When the field is weaker, e.g. and , it is a phase with . The spin configurations of the degenerate ground states of this phase are shown in Fig. 6. The two spins connected by become aligned due to the positive and the weak field .
iv.4 Phase III
If , the system is in a frustrated phase. We find four degenerate ground states in this phase contributing to the residual entropy . The other properties are given by , and . The spin configurations are shown in Fig. 7. In this case, the two spins connected by become antiparallel since is antiferromagnetic.
iv.5 Phase IV
In the case of and , phase III evolves into phase IV, which has and . Only one spin points down in this phase and it should be one of the two connected by . The spin configurations are shown in Fig. 8.
iv.6 Phase VII
Phase VII is a new phase when becomes positive in none-zero field. In this phase, and , the spins on the same triangle are parallel, however, antiparallel to the neighboring triangles for the positive and negative . If we treat the three spins on the same triangle as a higher spin located in the center of the triangle, it becomes an antiferromagnetic phase in a honeycomb lattice. This state gives and m=0.
iv.7 Phase diagram
V Monte Carlo Simulations
In this section we show the Monte Carlo (MC) simulation results of the star lattice Ising model with different combinations of parameters, which helps to corroborate our analytic predictions. Meanwhile, they allow us to calculate the magnetization and susceptibility at finite temperature.
We choose system size , , , where is the length of the unit cell for the star lattice, which means that the total number of spins is , as there are six spins in each unit cell. The periodic boundary condition is used for the simulations.
The specific heat and magnetic susceptibility during the MC simulations can be calculated using the fluctuation-dissipation theorem
where and are respectively the Monte Carlo averages of the total energy (i.e., the Hamiltonian) and magnetization.
Fig. 12 shows the temperature dependence of heat capacity per site at for a typical unfrustrated case , . The MC results are consistent with the exact analytic results.
We also calculate the susceptibility from the MC simulations, which is shown in Fig. 13. As increases, the peaks become sharper and sharper, indicating a phase transition.
We also study two different combinations of interactions for the unfrustrated case, which is , in Figs. 13 and 14. The temperature dependence of susceptibility is found to be sensitive to the sign of . When , the susceptibility has a sharp peak at the critical point between the ferromagnetic phase and paramagnetic phase. However, when , there is no such peak, which implies no phase transition in this case, just as what we get in exact solutions. The shape of the susceptibility peak depends on the size of the system when , whereas for , the size of the system has no influence on the susceptibility.
In summary, we have studied the Ising model on the star lattice with two different exchange couplings and using both analytical method and Monte Carlo simulations. We have presented its thermodynamic properties including internal energy, free energy, specific heat, entropy and susceptibility in the zero field. The phase transition temperature for is exactly same as the one found in Ref. Barry and Khatun (1995). There is no phase transition found if one of the couplings is antiferromagnetic. Moreover, we have obtained the rich phase diagrams in terms of and at zero temperature. Monte Carlo simulation is used to confirm the exact results and calculate the susceptibility.
In the fully frustrated case, the residual entropy of the system can be expressed as a closed form ( as showed in Eq. (24) which is consistent with the triangular and Kagome lattices. The system is less frustrated compared to the other triangulated lattices.
Our study provides a benchmark calculation for the thermodynamics of Ising spins on the star lattice, which can help experimentalists to investigate the real materials.
Acknowledgements.We thank Xiao-Ming Chen and Ming-Liang Tong for helpful discussions. This work is supported by the Fundamental Research Funds for the Central Universities of China (11lgjc12 and 10lgzd09), NSFC-11074310 and 11275279, MOST of China 973 program (2012CB821400), Specialized Research Fund for the Doctoral Program of Higher Education (20110171110026), Undergraduate Training Program at SYSU and NCET-11-0547.
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