Diamond-type Hierarchical Lattices Research Articles

Phase diagrams, quantum correlations and critical phenomena of antiferromagnetic Heisenberg model on diamond-type hierarchical lattices

The spin-1/2 antiferromagnetic (AF) Heisenberg systems are studied on three typical diamond-type hierarchical lattices (systems A, B and C) with fractal dimensions 1.63, 2 and 2.58, respectively, and the phase diagrams, critical phenomena and quantum correlations are calculated by a combination of the equivalent transformation and real-space renormalization group methods. We find that there exist a reentrant behavior for system A and a finite temperature phase transition in the isotropic Heisenberg limit for system C (not for system B). Unlike the ferromagnetic case, the Néel temperatures of AF systems A and B are inversely proportional to (when Δ → Δc) and ln Δ (when Δ → 0), respectively. And we also find that there is a turning point of quantum correlation in the isotropic Heisenberg limit Δ = 0 where there is a ‘peak’ of the contour and no matter how large the size of system is, quantum correlation will change to zero in the Ising limit for the three systems. The quantum correlation decreases with the increase of lattice size L and it is almost zero when L ⩾ 30 for system A, and for systems B and C, they still exist when L is larger than that of system A. Moreover, we discuss the effects of quantum fluctuation and analyze the errors of results in the above three systems, which are induced by the noncommutativity.

Robust thermal quantum correlation and quantum phase transition of spin system on fractal lattices

We investigate the quantum correlation measured by quantum discord (QD) for thermalized ferromagnetic Heisenberg spin systems in one-dimensional chains and on fractal lattices using the decimation renormalization group approach. It is found that the QD between two non-nearest-neighbor end spins exhibits some interesting behaviors which depend on the anisotropic parameter Δ, the temperature T, and the size of system L. With increasing Δ continuously, the QD possesses a cuspate change at Δ = 0 which is a critical point of quantum phase transition (QPT). There presents the “regrowth” tendency of QD with increasing T at Δ 0. As the size of the system L becomes large, there still exists considerable thermal QD between long-distance end sites in spin chains and on the fractal lattices even at unentangled states, and the long-distance QD can spotlight the presence of QPT. The robustness of QD on the diamond-type hierarchical lattices is stronger than that in spin chains and Koch curves, which indicates that the fractal can affect the behaviors of quantum correlation.

Thermal entanglement between non-nearest-neighbor spins on fractal lattices

We investigate the thermal entanglement between two end sites in spin chains and on fractal lattices by taking the negativity as a measure and using the decimation renormalization-group method. The effects of the temperature $T$, the anisotropy parameter $\ensuremath{\Delta}$, and the size of the system, $L$, on the entanglement are examined in detail. It is found that the entanglement decreases monotonically with increasing $T$ and vanishes beyond a critical value ${T}_{c}$. Our results also show that with increasing $\ensuremath{\Delta}$ from $\ensuremath{-}\ensuremath{\infty}$ to zero the entanglement first increases to the maximum and then decreases sharply to zero. Different from the cases of spin chains and Koch curves, the entanglement on the diamond-type hierarchical (DH) lattices presents some interesting behaviors. As the sizes of the DH lattices, $L$, become large, the entanglement is rather robust and there exists sizable entanglement between long-distance end sites. This result indicates that different fractal structures can result in various entanglement properties.

Critical properties of the S4 model on a special diamond-type hierarchical lattice

Using the renormalization-group transformation and cumulative expansion technique, the phase transition and critical properties of the S4 model on a special diamond-type hierarchical lattice are studied, and its fixed points and critical exponents are obtained. The results show that there exists a Wilson-Fisher fixed point besides the Gaussian fixed point, and compared with the Gaussian model of the special diamond-type hierarchical lattices, the critical exponents have changed.

Critical properties of the model on diamond-type hierarchial lattices

We study the critical properties of the S-4 model on diamond-type hierarchical lattices in the presence of an external magnetic field. It is assumed that for this type of inhomogenous fractal lattice, the Gaussian distribution constant, the four-spins interaction parameter and the external magnetic field in the S4 model depend on the coordination number of the site on the fractal lattices. By combining the real-space renormalization-group scheme with the cumulative expansion method, we obtain the critical points and further calculate critical exponents according to the scaling theory. The results show that on diamond-type hierarchical lattices with branches m > 4 of 2 bonds, the critical point of the S4 model is just the Gaussian fixed point, and therefore critical exponents are in full agreement with those of the Gaussian model, and that on those with m <= 4, the Wilson-Fisher fixed point as well as the Gaussian fixed point is obtained. The Wilson-Fisher fixed point has a decisive influence on the critical behavior of the system, and critical exponents are related to the fractal dimensionality. We found that the S4 model on the fractal lattices and that on the translation symmetric lattices show similar behaviors in the dependence of the critical properties on the dimensionality. (c) 2005 Elsevier B.V. All rights reserved.

Critical dynamics of the kinetic Glauber-Ising model on hierarchical lattices.

The critical dynamics of the kinetic Glauber-Ising model is studied on a family of the diamond-type hierarchical lattices with various branches. By carrying out the time-dependent real-space renormalization-group transformation to the master equation of the systems considered, the dynamic exponent is calculated. We find that the dynamic exponent depends on fractal dimension d(f) or the branch number m in a generator, and that it increases with the increase of d(f) or m. We notice that for the case of m=1 (one-dimensional spin chain, d(f)=1) our result z=2 is the same as the exact result obtained by Glauber, and for the case of m=2 (the simplest one in the diamond-type hierarchical lattices, d(f)=2) the exponent z=2.626 is higher than those of the two-dimensional regular lattice and the triangular lattice.

Phase transitions of the anisotropic Ashkin-Teller model on a family of diamond-type hierarchical lattices.

The phase transitions of the anisotropic Ashkin-Teller model on a family of diamond-type hierarchical lattices is studied by means of the transfer-matrix method and the real-space renormalization-group transformation. We find that the phase diagram, for the ferromagnetic case, consists of five phases, i.e., the fully disordered paramagnetic phase P, the fully ordered ferromagnetic phase F, and three partially ordered ferromagnetic phases F(s), F(sigma), and F(s sigma), as well as ten nontrivial fixed points. The correlation length critical exponents and the crossover exponents are also calculated. In addition, we also investigate the variations of the critical exponents with the fractal dimension d(f), the number of branches m, and the number of bonds per branch b of the generator of the family of diamond-type hierarchical lattices. Finally we give a brief discussion about universality.

Critical slowing down of the Gaussian spin system on diamond-type hierarchical lattices

Based on the single-spin transition critical dynamics, we have investigated the critical slowing down of the Gaussian spin model situated on the fractal family of diamond-type hierarchical lattices. We calculate the dynamical critical exponent z and the correlation-length critical exponent ν using the dynamical decimation renormalization-group technique. The result, together with some earlier ones, suggests us to conclude that on a wide range of geometries, zν = 1 is the general relationship, while the two exponents depend on the specific structure. However, we have investigated for various lattices in an earlier paper, the system studied in this paper shows highly universal z = 1/ν = 2 independent of the structure and the dimensionality.

Critical Behavior of the Gaussian Model with Periodic Interactions on Diamond-Type Hierarchical Lattices in External Magnetic Fields**The project supported by the National Basic Research Project “Nonlinear Science”, National Natural Science Foundation of China, the Zhejiang Provincial “151 Talent Engineering” Foundation and the Wenzhou Municipal “551 Talent Engineering” Foundation of China

The Gaussian spin model with periodic interactions on the diamond-type hierarchical lattices is constructed by generalizing that with uniform interactions on translationally invariant lattices according to a class of substitution sequences. The Gaussian distribution constants and imposed external magnetic fields are also periodic depending on the periodic characteristic of the interaction bonds. The critical behaviors of this generalized Gaussian model in external magnetic fields are studied by the exact renormalization-group approach and spin rescaling method. The critical points and all the critical exponents are obtained. The critical behaviors are found to be determined by the Gaussian distribution constants and the fractal dimensions of the lattices. When all the Gaussian distribution constants are the same, the dependence of the critical exponents on the dimensions of the lattices is the same as that of the Gaussian model with uniform interactions on translationally invariant lattices.

Critical behavior of the Gaussian model on fractal lattices in external magnetic field

For inhomogeneous lattices we generalize the classical Gaussian model, i.e. it is proposed that the Gaussian type distribution constant and the external magnetic field of site i in this model depend on the coordination number q i of site i , and that the relation b q i /b q j =q i q j holds among b q i 's, where b q i is the Gaussian type distribution constant of site i . Using the decimation real-space renormalization group following the spin-rescaling method, the critical points and critical exponents of the Gaussian model are calculated on some Koch type curves and a family of the diamond-type hierarchical (or DH) lattices. At the critical points, it is found that the nearest-neighbor interaction and the magnetic field of site i can be expressed in the form K * = b q i / q i and h q 1 * =0, respectively. It is also found that most critical exponents depend on the fractal dimensionality of a fractal system. For the family of the DH lattices, the results are identical with the exact results on translation symmetric lattices, and if the fractal dimensionality d f =4, the Gaussian model and the mean field theories give the same results.

Critical behavior of Gaussian model on diamond-type hierarchical lattices

It is proposed that the Gaussian type distribution constantb qi in the Gaussian model depends on the coordination numberq i of sitei, and that the relation $${{b_{q_i } } \mathord{\left/ {\vphantom {{b_{q_i } } {b_{q_j } }}} \right. \kern-\nulldelimiterspace} {b_{q_j } }} = {{q_q } \mathord{\left/ {\vphantom {{q_q } {q_q }}} \right. \kern-\nulldelimiterspace} {q_q }}$$ holds amongb qi ’s. The Gaussian model is then studied on a family of the diamond-type hierarchical (or DH) lattices, by the decimation real-space renormalization group following spin-rescaling method. It is found that the magnetic property of the Gaussian model belongs to the same universal class, and that the critical pointK* and the critical exponentv are given by $${{K^ * = b_{q_i } } \mathord{\left/ {\vphantom {{K^ * = b_{q_i } } {q_i }}} \right. \kern-\nulldelimiterspace} {q_i }}$$ and $$\nu = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}$$ , respectively.

Critical behavior and the Harris criterion for the random Potts model on hierarchical lattices.

The critical properties of the diluted Potts model on diamond-type hierarchical lattices of different fractal dimensions were studied recently by Wu et al. [Phys. Rev. B 48, 3171 (1993)]. Although the aim of the paper is very interesting, we show in this comment that the fixed bimodal bond-probability distribution used in their renormalization-group work is not appropriate to deal with this disordered model. Such treatment violates the Harris criterion and may lead to spurious results. We point out that the correct way to deal with randomness is to rescale the full distribution and take its second moment as an additional parameter. We use this method and find that the critical properties are different from those which Wu et al. predicted.

Self-avoiding random walks on a family of diamond-type hierarchical lattices.

We have used the exact renormalization-group method proposed by Dhar to study self-avoiding random walks (SAW's) on a family of hierarchical lattices. The generator of the lattices is made of l branches and each branch has m bonds. We expect that since the lattices are infinitely ramified, the critical exponents of SAW's should be different from that on finitely ramified lattices and belong to a new universal class. We calculated the critical exponents \ensuremath{\alpha}, \ensuremath{\nu}, and \ensuremath{\gamma} under the condition lm, and, with ${\mathit{D}}_{\mathit{f}}$ the fractal dimension, obtained the scaling law ${\mathit{D}}_{\mathit{f}}$\ensuremath{\nu}=2-\ensuremath{\alpha}, which agrees with other authors's conclusions. When l\ensuremath{\ge}m, we cannot work out the problem, and some discussion is given.

Diluted Potts model on diamond-type hierarchical lattices.

The bond-diluted q-state Potts model situated on a fractal family of diamond-type hierarchical lattices is investigated with the renormalization-group method. We find that in the ferromagnetic-interaction case there is a borderline value ${\mathit{q}}_{\mathit{t}}$ on the lattice with ${\mathit{D}}_{\mathit{f}}$\ensuremath{\le}2. When q&gt;${\mathit{q}}_{\mathit{t}}$ there is a crossover to a diluted fixed point (critical region). However, two borderline values ${\mathit{q}}_{\mathit{t}1}$ and ${\mathit{q}}_{\mathit{t}2}$ (${\mathit{q}}_{\mathit{t}1}$${\mathit{q}}_{\mathit{t}2}$) appear on the lattices with ${\mathit{D}}_{\mathit{f}}$&gt;2. In the region ${\mathit{q}}_{\mathit{t}1}$q${\mathit{q}}_{\mathit{t}2}$, the two diluted fixed points coexist, one is critical and another is tricritical. These fixed points merge and annihilate at ${\mathit{q}}_{\mathit{t}1}$. As q increases from ${\mathit{q}}_{\mathit{t}1}$, the tricritical point approaches and meets with the pure fixed point at ${\mathit{q}}_{\mathit{t}2}$. The critical behavior of the system is governed by the critical diluted fixed point when the concentration of occupied bonds is below its tricritical value, otherwise it is governed by the pure fixed point. This feature implies that the concentration of occupied bonds is relevant to the critical properties of the system. We also discuss the antiferromagnetic case and our results show that the critical properties of the system are unchanged by dilution.

Diamond-type hierarchical lattices for the Potts antiferromagnet.

The Potts antiferromagnet situated on the fractal family of diamond-type hierarchical lattices is investigated. The exact analysis of our findings reveals that the algebraically ordered behavior predicted by Berker and Kadanoff can exist on such lattices with L=odd, where L is the structure parameter of the hierarchical lattices; however, similar behavior to that of the antiferromagnetic Potts model on a decorated square lattice is exhibited for systems with L=even. We find that the different constructions that have the same fractal dimensionality can have a different cutoff value ${\mathit{q}}_{0}$, in contrast with the translationally invariant system. We also give the ground-state entropy at q=${\mathit{q}}_{0}$ in a closed form for the systems with L=even.

Family of diamond-type hierarchical lattices.

A family of the diamond-type hierarchical lattices as a kind of fractals is proposed, on which the Ising model is exactly solved. The convergent condition of the free energy per bond in the thermodynamic limit is obviously given. We find that the unstable fixed point of the renormalization-group transformation moves toward K=0 (T=\ensuremath{\infty}) from K=\ensuremath{\infty} (T=0) as the number of branches P is increased, and when P=\ensuremath{\infty} (i.e., ${d}_{f}$=\ensuremath{\infty}) the unstable fixed point K=0 exactly, in contrast with that of the Bethe lattice. We also calculate the critical exponent of the correlation length; we find that in the ${d}_{f}$=2 and 3 cases the exponent is different from that of the regular lattice with d=2 and 3, respectively, which seems to imply that more general criteria for the classification of universality should be proposed. We have also discussed the upper critical fractal dimension.