Triangular Ising antiferromagnet in a staggered field

Dhar, Abhishek; Chaudhuri, Pinaki; Dasgupta, Chandan

doi:10.1103/physrevb.61.6227

Cited by 24 publications

(33 citation statements)

References 19 publications

(16 reference statements)

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“…Like these glassy systems, the triangular Ising antiferromagnet has frustration and an exponentially large number of ground states. Dhar et al [10] have studied the equilibrium properties of a triangular Ising antiferromagnet in the presence of an ordering field, which is conjugate to one of the degenerate ground states using Monte Carlo simulations with three different dynamics. They found, in all the three cases, that equilibrium times for low-field values increase rapidly with system size at low temperature, so these results do not permit a definite conclusion about the existence of a phase transition for finite J, while they prove the existence of a phase transition if J is infinitely large.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Ising spins with antiferromagnetic bonds on a triangular lattice

Ismail¹,

Salem

2003

Physica Status Solidi (b)

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Six-coupled Ising spins on a triangular lattice with antiferromagnetic (AF) nearest neighbour and ferromagnetic (F) next-nearest neighbour interactions are investigated by Glauber dynamics. The system is fully frustrated and has seven non-trivial local energy minima in both clusters. The ground state has four degenerate states with energy -6. These states are separated by the energy barrier ∆E = 4.0 to invert spins in the ground state. The dynamics of AF-F and F-AF coupling clusters are solved exactly. Each cluster contributes as many long relaxation times as it has non-trivial local energy minima. The barriers against inversion of the clusters take only two values, 0 and 4|J|. In the paramagnetic case (PM), there is no diverging relaxation time. In the ferromagnetic case (FM), there is only one long-lived mode. The longest relaxation times follow an Arrhenius law. Both AF-F and F-AF clusters undergo a zero-temperature phase transition. The real and imaginary parts of the dynamic susceptibility display maxima if plotted versus temperature. The frequency dependence of the susceptibilities explain the effect of frustration. The real part reveals two plateaus and the imaginary part displays two corresponding maxima if they are plotted as functions of the logarithm of frequency. The Argand plots show two overlapping semicircles for fixed frequency at low temperature, indicating the time separation of the modes.

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Section: Introductionmentioning

confidence: 99%

Dynamics of Ising spins with antiferromagnetic bonds on a triangular lattice

Ismail¹,

Salem

2003

Physica Status Solidi (b)

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“…This feature is reminiscent of classical lattice models, which have ground state degeneracy that leads to finite entropy. In these models, the subextensivity of the conserved quantity has nontrivial implications for phase transitions [36]. It would be interesting to explore a theory of grains based on the conservation of ( F x , F y ).…”

Section: Conservation Principlesmentioning

confidence: 99%

The Statistical Physics of Athermal Materials

Henkes

Daniels

et al. 2015

Annu. Rev. Condens. Matter Phys.

138

140

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At the core of equilibrium statistical mechanics lies the notion of statistical ensembles: a collection of microstates, each occurring with a given a priori probability that depends only on a few macroscopic parameters such as temperature, pressure, volume, and energy. In this review article, we discuss recent advances in establishing statistical ensembles for athermal materials. The broad class of granular and particulate materials is immune from the effects of thermal fluctuations because the constituents are macroscopic. In addition, interactions between grains are frictional and dissipative, which invalidates the fundamental postulates of equilibrium statistical mechanics. However, granular materials exhibit distributions of microscopic quantities that are reproducible and often depend on only a few macroscopic parameters. We explore the history of statistical ensemble ideas in the context of granular materials, clarify the nature of such ensembles and their foundational principles, highlight advances in testing key ideas, and discuss applications of ensembles to analyze the collective behavior of granular materials.

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“…The order parameter associated with this entropy-vanishing transition is the density of extended string-like structures which characterize the TIAFM ground states. The string picture of the TIAFM ground states derives from a well-known mapping of these states to dimer coverings [13,14]. In a ground state, there is one unsatisfied bond per triangular plaquette and the dimers are the filled-in bonds of the dual honeycomb lattice that cross the unsatisfied bonds of the triangular lattice, as shown in Fig.(1).…”

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confidence: 99%

“…In a ground state, there is one unsatisfied bond per triangular plaquette and the dimers are the filled-in bonds of the dual honeycomb lattice that cross the unsatisfied bonds of the triangular lattice, as shown in Fig.(1). Superposing a dimer configuration on a "standard" dimer configuration where all the dimers are vertical [13,14] leads to a string configuration (cf Fig.(1)). Assuming spin-flip dynamics for the moment, the only spins that can be changed while the system remains in the ground-state manifold, are the ones which have a coordination of 3-3 (3 satisfied and 3 unsatisfied bonds).…”

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confidence: 99%

“…Exact results We can solve the CTIAFM exactly within the restricted spin ensemble of the ground states of the TIAFM. All the states in this ensemble can be classified according to the string density p = N s /L where L is the linear dimension of the sample and N s is the number of strings [13]. The number of spin states (Ω(p)) belonging to a particular string-density sector p has been shown to be exponentially large [13]: Ω(p) = exp(N γ(p)), with N = L × L being the total number of spins.…”

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Entropy-Vanishing Transition and Glassy Dynamics in Frustrated Spins

Chakraborty

2001

Phys. Rev. Lett.

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In an effort to understand the glass transition, the dynamics of a non-randomly frustrated spin model has been analyzed. The phenomenology of the spin model is similar to that of a supercooled liquid undergoing the glass transition. The slow dynamics can be associated with the presence of extended string-like structures which demarcate regions of fast spin flips. An entropy-vanishing transition, with the string density as the order parameter, is related to the observed glass transition in the spin model.The glass transition in supercooled liquids is heralded by anomalously slow relaxations with a time scale diverging as the liquid freezes into the glassy state [1]. In recent years, there have been careful experimental and theoretical studies aimed at understanding the structural aspects of this transition. The presence and nature of dynamical heterogeneities near the glass transition has been a dominant underlying theme of both simulations [2,3] and experiments [4,5]. Simulations in Lennard-Jones liquids [2] have shown the existence of string-like dynamical heterogeneities and similar structures have been observed directly in colloidal glasses [5]. In the Adam-Gibbs scenario, the glass transition is related to a phase transition accompanied by the vanishing of configurational entropy [6,7]. An explicit connection between (a) dynamical heterogeneities, (b) anomalous relaxations and (c) the Adam-Gibbs scenario would provide useful insight into the nature of the glass transition. In this work, we present our analysis of a simple model where there are naturally occurring dynamical heterogeneities in the form of strings and where there is an entropy vanishing transition involving these structures. Monte Carlo simulations of the model show that there is a glass-like transition with diverging time scales and an anomalously broad relaxation spectrum. Analysis of the simulation results provides strong evidence that the entropy-vanishing transition underlies the observed dynamical behavior.a. Model One of the simplest non-randomly frustrated spin models is the triangular-lattice Ising antiferromagnet (TIAFM). The TIAFM has an exponentially large number of ground states and has a zerotemperature critical point [8][9][10]. The model studied in this letter is the compressible TIAFM (CTIAFM) in which the coupling of the spins to the elastic strain fields removes the exponential degeneracy of the groundstate. We solve the CTIAFM exactly within the groundstate ensemble of the TIAFM and show that there is an entropy-vanishing transition which involves extended structures. We then present results of simulations which indicate that the entropy-vanishing transition leads to glassy dynamics.The Hamiltonian of the CTIAFM is:Here J, the strength of the anti-ferromagnetic coupling, is modulated by the presence of the second term which defines a coupling between the spins and the homogeneous strain fields e α , α = 1, 2, 3, along the three nearest-neighbor directions on the triangular lattice. The last term stabilizes the unstrained lat...

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Triangular Ising antiferromagnet in a staggered field

Cited by 24 publications

References 19 publications

Dynamics of Ising spins with antiferromagnetic bonds on a triangular lattice

Dynamics of Ising spins with antiferromagnetic bonds on a triangular lattice

The Statistical Physics of Athermal Materials

Entropy-Vanishing Transition and Glassy Dynamics in Frustrated Spins

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