Spin systems on hierarchical lattices. II. Some examples of soluble models

Kaufman, Miron; Griffiths, Robert B.

doi:10.1103/physrevb.30.244

Cited by 128 publications

(86 citation statements)

References 42 publications

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“…[20,21,22] The renormalization group analysis on the hierarchical lattice is an exact technique to obtain the location of the transition point, though it is difficult to obtain such an exact solution on regular lattices.…”

Section: Hierarchical Latticementioning

confidence: 99%

Multicritical points for spin-glass models on hierarchical lattices

2008

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The locations of multicritical points on many hierarchical lattices are numerically investigated by the renormalization group analysis. The results are compared with an analytical conjecture derived by using the duality, the gauge symmetry and the replica method. We find that the conjecture does not give the exact answer but leads to locations slightly away from the numerically reliable data. We propose an improved conjecture to give more precise predictions of the multicritical points than the conventional one. This improvement is inspired by a new point of view coming from renormalization group and succeeds in deriving very consistent answers with many numerical data.

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Section: Hierarchical Latticementioning

confidence: 99%

Multicritical points for spin-glass models on hierarchical lattices

2008

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“…A van der Waals loop [13] develops in the stress-strain dependence signaling a weak phase transition that replaces the continuous Potts phase transition. While in the renormalization-group calculations [14,15] that are exact [16] on hierarchical lattices [17,18] the Potts transitions are always continuous, in the Monte Carlo simulations we can see both continuous transitions (for small q) and discontinuous transitions (for large q). This allows us to explore the influence of the order of the Potts transition on the mechanical properties of the solid, such as stress dependence on temperature and interatomic distance.…”

Section: Introductionmentioning

confidence: 97%

“…The recursion equations (16)- (18) represent the exact solutions [16] for the diamond hierarchical lattice [17,18]. of those quantities is scaled by the total number of lattice bonds.…”

Section: Renormalization Groupmentioning

confidence: 99%

Equation of state from the Potts-percolation model of a solid

Kaufman

Diep

2011

Phys. Rev. E

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We expand the Potts-percolation model of a solid to include stress and strain.Neighboring atoms are connected by bonds. We set the energy of a bond to be given by the Lennard-Jones potential. If the energy is larger than a threshold the bond is more likely to fail, while if the energy is lower than the threshold the bond is more likely to be alive. In two dimensions we compute the equation of state: stress as function of inter-atomic distance and temperature by using renormalization group and Monte Carlo simulations. The phase diagram, the equation of state, and the isothermal modulus are determined. When the Potts heat capacity is divergent the continuous transition is replaced by a weak first-order transition through the van der Waals loop mechanism. When the Potts transition is first order the stress exhibits a large discontinuity as function of the inter-atomic distance.

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“…Our method, previously described in extensive detail [6] and used on a qualitatively different model with qualitatively different results, is simultaneously the Migdal-Kadanoff approximation [15,16] for the cubic lattice and the exact solution [17][18][19][20][21] for a d = 3 hierarchical lattice, with length rescaling factor b = 3. Exact calculations on hierarchical lattices are also currently widely used in a variety of statistical mechanics [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], finance [37], and, most recently, DNA-binding [38] problems.…”

Section: Renormalization-group Method: Migdal-kadanoff Approximamentioning

confidence: 99%

“…Since bond moving in the Migdal-Kadanoff approximation [15,16] is done transversely to the bond directions, this sequencing is respected. Equivalently, in the corresponding hierarchical lattice [17][18][19][20][21], one can always define a direction along the connectivity, for example, from left to right, and assign consecutive increasing number labels to the sites. In Eq.…”

Section: Double Spin-glass System: Left-right Chiral and Ferro-anmentioning

confidence: 99%

Phase transitions between different spin-glass phases and between different chaoses in quenched random chiral systems

Çağlar

Berker

2017

Phys. Rev. E

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The left-right chiral and ferromagnetic-antiferromagnetic double-spin-glass clock model, with the crucially even number of states q = 4 and in three dimensions d = 3, has been studied by renormalization-group theory. We find, for the first time to our knowledge, four spin-glass phases, including conventional, chiral, and quadrupolar spin-glass phases, and phase transitions between spin-glass phases. The chaoses, in the different spin-glass phases and in the phase transitions of the spin-glass phases with the other spin-glass phases, with the non-spin-glass ordered phases, and with the disordered phase, are determined and quantified by Lyapunov exponents. It is seen that the chiral spin-glass phase is the most chaotic spin-glass phase. The calculated phase diagram is also otherwise very rich, including regular and temperature-inverted devil's staircases and reentrances.

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Spin systems on hierarchical lattices. II. Some examples of soluble models

Cited by 128 publications

References 42 publications

Multicritical points for spin-glass models on hierarchical lattices

Multicritical points for spin-glass models on hierarchical lattices

Equation of state from the Potts-percolation model of a solid

Phase transitions between different spin-glass phases and between different chaoses in quenched random chiral systems

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