The Blume–Capel model on hierarchical lattices: Exact local properties

Rocha-Neto, Mário J.G.; Camelo-Neto, G.; Nogueira, Eliane Maria de Souza; Coutinho, S.

doi:10.1016/j.physa.2017.11.156

Cited by 22 publications

(5 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1, the renormalization-group recursion relations of the Migdal-Kadanoff approximation are identical to those of an exact solution of a hierarchical lattice [7][8][9]. For recent works using hierarchical lattices, see [10][11][12][13][14][15][16][17][18][19][20][21][22] This simple renormalization-group transformation has been widely successful on different systems: the lowercritical dimension d c below which no ordering occurs has been correctly determined as d c = 1 for the Ising model [5,6], d c = 2 for the XY [23,24] and Heisenberg [25] models, and the presence of an algebraically ordered phase has been seen for the XY model [18,23,24]. In q-state Potts models, the number of states q c for the changeover from second-order to first-order phase transitions has been correctly obtained in d = 2 and 3.…”

Section: The Model and The General Methodsmentioning

confidence: 99%

Nematic Ordering in the Heisenberg Spin-Glass System in d=3 Dimensions

Tunca¹,

Berker²

2022

Preprint

View full text Add to dashboard Cite

Nematic ordering, where the spins globally align along a spontaneously chosen axis irrespective of direction, occurs in spin-glass systems of classical Heisenberg spins in d = 3. In this system where the nearest-neighbor interactions are quenched randomly ferromagnetic or antiferromagnetic, instead of the locally randomly ordered spin-glass phase, the system orders globally as a nematic phase. The system is solved exactly on a hierarchical lattice and, equivalently, Migdal-Kadanoff approximately on a cubic lattice. The global phase diagram is calculated, exhibiting this nematic phase, and ferromagnetic, antiferromagnetic, disordered phases. The nematic phase of the classical Heisenberg spin-glass system is also found in other dimensions d > 2: We calculate nematic transition temperatures in 24 dimensions in 2 < d ≤ 4.

show abstract

Section: The Model and The General Methodsmentioning

confidence: 99%

Nematic Ordering in the Heisenberg Spin-Glass System in d=3 Dimensions

Tunca¹,

Berker²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Recent works using exactly soluble hierarchical models are in Refs. [39][40][41][42][43][44][45][46][47].…”

Section: Method: Global Renormalization-group Theory Of Quenched Prob...mentioning

confidence: 99%

Phase Transitions of the Variety of Random-Field Potts Models

Turkoglu,

Berker

2021

Preprint

View full text Add to dashboard Cite

The phase transitions of random-field q-state Potts models in d = 3 dimensions are studied by renormalization-group theory by exact solution of a hierarchical lattice and, equivalently, approximate Migdal-Kadanoff solutions of a cubic lattice. The recursion, under rescaling, of coupled random-field and random-bond (induced under rescaling by random fields) coupled probability distributions is followed to obtain phase diagrams. Unlike the Ising model (q = 2), several types of random fields can be defined for q ≥ 3 Potts models, including random-axis favored, random-axis disfavored, random-axis randomly favored or disfavored cases, all of which are studied. Quantitatively very similar phase diagrams are obtained, for a given q for the three types of field randomness, with the low-temperature ordered phase persisting, increasingly as temperature is lowered, up to random-field threshold in d = 3, which is calculated for all temperatures below the zero-field critical temperature. Phase diagrams thus obtained are compared as a function of q. The ordered phase in the low-q models reaches higher temperatures, while in the high-q models it reaches higher random fields. This renormalization-group calculation result is physically explained.

show abstract

“…Since the work of Silva et al [44], the WL algorithm has been used extensively in the BC model to extract precise data of thermodynamic properties in both the first-order and continuous phase transition boundaries [20,27,[45][46][47][48]. However, one should notice that these studies are mostly restricted to regular lattices, namely, the square, triangular, and simple-cubic lattices, with the exception of a few works on small-world networks without crystal field [49,50] and a fractal structure with a scale-free degree distribution [51]. In general, the role of lattice structure on the phase diagram of the BC model has been overlooked.…”

Section: Introductionmentioning

confidence: 99%

Monte Carlo studies of the Blume–Capel model on nonregular two- and three-dimensional lattices: phase diagrams, tricriticality, and critical exponents

Azhari¹,

Yu²

2022

J. Stat. Mech.

View full text Add to dashboard Cite

We perform Monte Carlo simulations, combining both the Wang–Landau and the Metropolis algorithms, to investigate the phase diagrams of the Blume–Capel model on different types of nonregular lattices (Lieb lattice (LL), decorated triangular lattice (DTL), and decorated simple cubic lattice (DSC)). The nonregular character of the lattices induces a double transition (reentrant behavior) in the region of the phase diagram at which the nature of the phase transition changes from first-order to second-order. A physical mechanism underlying this reentrance is proposed. The large-scale Monte Carlo simulations are performed with the finite-size scaling analysis to compute the critical exponents and the critical Binder cumulant for three different values of the anisotropy Δ / J ∈ 0 , 1 , 1.34 ( for LL ) , 1.51 ( for DTL and DSC ) , showing thus no deviation from the standard Ising universality class in two and three dimensions. We report also the location of the tricritical point to considerable precision: (Δ t /J = 1.3457(1); k B T t /J = 0.309(2)), (Δ t /J = 1.5766(1); k B T t /J = 0.481(2)), and (Δ t /J = 1.5933(1); k B T t /J = 0.569(4)) for LL, DTL, and DSC, respectively.

show abstract

The Blume–Capel model on hierarchical lattices: Exact local properties

Cited by 22 publications

References 48 publications

Nematic Ordering in the Heisenberg Spin-Glass System in d=3 Dimensions

Nematic Ordering in the Heisenberg Spin-Glass System in d=3 Dimensions

Phase Transitions of the Variety of Random-Field Potts Models

Monte Carlo studies of the Blume–Capel model on nonregular two- and three-dimensional lattices: phase diagrams, tricriticality, and critical exponents

Contact Info

Product

Resources

About