Classical dimers with aligning interactions on the square lattice

Alet, Fabien; Ikhlef, Yacine; Jacobsen, Jesper Lykke; Misguich, Grégoire; Pasquier, Vincent

doi:10.1103/physreve.74.041124

Cited by 105 publications

(252 citation statements)

References 58 publications

(113 reference statements)

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“…In order to do this we follow in spirit Ref. [38][39][40] in forming a directed loop or "worm" algorithm. First we will consider guiding the loop formation using the J 2 and J 3 interactions, and then later we will briefly mention how to also include the J 4 and J 5 interactions.…”

Section: Figmentioning

confidence: 99%

“…In order to perform Monte Carlo simulations of the triangular lattice Ising antiferromagnet (TLIAF) with J 1 → ∞ we have employed a worm algorithm [38][39][40] . This allows the system to make transitions between different winding number sectors and is designed such that proposed Monte Carlo updates are always accepted.…”

Section: Appendix B: Worm Algorithm For Monte Carlo Updatesmentioning

confidence: 99%

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Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets

2016

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Using a specially designed Monte Carlo algorithm with directed loops, we investigate the triangular lattice Ising antiferromagnet with coupling beyond nearest neighbour. We show that the first-order transition from the stripe state to the paramagnet can be split, giving rise to an intermediate nematic phase in which algebraic correlations coexist with a broken symmetry. Furthermore, we demonstrate the emergence of several properties of a more topological nature such as fractional edge excitations in the stripe state, the proliferation of double domain walls in the nematic phase, and the Kasteleyn transition between them. Experimental implications are briefly discussed.The triangular-lattice Ising antiferromagnet (TLIAF) is the archetypal model of frustration. Ground states of the nearest-neighbour (n.n.) model obey the local constraint that triangles cannot host three equivalent Ising spins, and it follows that there is an extensive entropy 1,2 . This results in a critical state, characterised by algebraic correlations between the spins 3,4 .In reality, interactions are rarely limited to n.n., and a more realistic Hamiltonian takes the form,where σ i = ±1 and J ij > 0. This model is experimentally relevant in a diverse range of systems, including artificial dipolar magnets 5 , materials such as Ba 3 CuSb 2 O 9 where electrically charged dumbbells act as Ising degrees of freedom 6,7 , trapped ions 8,9 , frustrated Coulomb liquids 10 , Josephson junction arrays 11 and absorbed monolayers 12 .In spite of its ubiquity, this model has received limited attention. The difficulty in analysing H Is [Eq. 1] arises from the critical nature of the n.n. ground-state manifold, which is very sensitive to perturbation, and in the presence of further-neighbour coupling the model is not amenable to an analytic solution. Besides, in the limit J 1 → ∞ as compared to the other characteristic energy scales of the problem (J 2 , J 3 , etc.), Monte Carlo (MC) simulations based on the Metropolis algorithm are unable to reach the ground state, and the problem of freezing remains even when this constraint is relaxed, for example in the case of dipolar interactions 13 .The current understanding of the properties of H Is is based on estimates of the energy and entropy of different types of extended defects. This results in the prediction that the broken Z 2 ×Z 3 symmetry of the low-temperature stripe state 14,16,17 can be restored either in a single firstorder transition, or via a pair of transitions, where the low-temperature, Z 2 -restoring transition is second order and the higher-temperature, Z 3 -restoring transition is * deceased FIG. 1:Representative phase diagrams of the TLIAF with J 1 → ∞, determined by MC simulation. All phase boundaries were determined from the winding number. first order 14 . When the transition is split, an intermediate phase of nematic type is revealed, and it is characterised by a set of fluctuating double domain walls 14 (ddw).In this letter, we show that the difficulty in simulating the TLIAF with MC arises...

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Section: Figmentioning

confidence: 99%

Section: Appendix B: Worm Algorithm For Monte Carlo Updatesmentioning

confidence: 99%

Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets

2016

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“…(2.1) by means of a Monte Carlo worm algorithm with a local heat-bath detailed balance condition [23]. We have set the energy scale to be v = 1 > 0.…”

Section: Results From Monte Carlo Simulationsmentioning

confidence: 99%

“…First, we show that the key ingredient for stabilizing nematic dimer liquids is a strong same-row or same-column dimer-aligning, spatially isotropic, interaction. In a fully packed lattice this interaction leads to crystalline phases [4,23,[34][35][36]. Here we will show that at finite hole density it stabilizes an Ising nematic fluid phase.…”

Section: Introductionmentioning

confidence: 95%

Ising nematic fluid phase of hard-core dimers on the square lattice

2014

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We present a model of classical hard-core dimers on the square lattice that contains an Ising nematic phase in its phase diagram. We consider a model with an attractive interaction for parallel dimers on a given plaquette of the square lattice and an attractive interaction for neighboring parallel dimers on the same row ({\it viz} column) of the lattice. By extensive Monte carlo simulations we find that with a finite density of holes the phase diagram has, with rising temperatures, a columnar crystalline phase, an Ising nematic liquid phase and a disordered fluid phase, separated by Ising continuous phase transitions. We present strong evidence for the Ising universality class of both transitions. The Ising nematic phase can be interpreted as either an intermediate classical thermodynamic phase (possibly of a strongly correlated antiferromagnet) or as a phase of a 2D quantum dimer model using the Rokhsar-Kivelson construction of exactly solvable quantum Hamiltonians.Comment: 13 pages, 24 figure

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“…11 are not ''ideal'' (3) because there is a small energetic penalty for the parallel tile arrangement. They are ''interacting'' random tilings (15)(16)(17)(18). This difference in energy is however below the critical value at which such tilings would undergo a Kosterlitz-Thouless (KT) transition to an ordered phase (18).…”

Section: Modelmentioning

confidence: 99%

Molecular random tilings as glasses

Garrahan

Stannard

Blunt

et al. 2009

Proc. Natl. Acad. Sci. U.S.A.

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We have recently shown that p-terphenyl-3,5,3 ,5 -tetracarboxylic acid adsorbed on graphite self-assembles into a two-dimensional rhombus random tiling. This tiling is close to ideal, displaying long-range correlations punctuated by sparse localized tiling defects. In this article we explore the analogy between dynamic arrest in this type of random tilings and that of structural glasses. We show that the structural relaxation of these systems is via the propagation-reaction of tiling defects, giving rise to dynamic heterogeneity. We study the scaling properties of the dynamics and discuss connections with kinetically constrained models of glasses.dimer coverings ͉ dynamic heterogeneity ͉ glass transition ͉ lozenge tilings D imer coverings of lattices and random polygon tilings are problems of great interest both in physics and mathematics (1)(2)(3)(4)(5). One of the reasons is that they correspond to the idealized description of systems that display entropically stabilized critical phases and fractional excitations, being therefore relevant to condensed-matter systems such as quasicrystals (6, 7) and frustrated antiferromagnets (8-10).However, actual experimental realizations of dimer covering/ random tiling systems are very rare. Fig. 1 shows one of them: a molecular network formed by organic molecules adsorbed from solution onto a graphite substrate (11). The molecule p-terphenyl-3,5,3Ј,5Ј-tetracarboxylic acid (TPTC) (see Fig. 1 A) binds to other TPTC molecules on the substrate adopting one of three possible orientations. Each molecule can then be mapped to a rhombus tile (see Fig. 1B), where the colors red, green, and blue indicate the molecular orientation. Neighboring molecules (tiles) can bind to neighbors in a parallel or ''arrowhead'' configuration, equivalent to junctions between tiles of the same or different color, respectively. Fig. 1C shows a scanning tunneling microscope (STM) image of the resulting molecular network of adsorbed TPTC, and Fig. 1D shows the corresponding rhombus tiling (11), where each molecule is represented by a tile.The molecular networks studied in ref. 11 are close to ''perfect'' rhombus tilings (or dimer coverings of the honeycomb lattice) (1)(2)(3)8) in the sense that they contain rather few tiling defects, typically less than one defect per 300 adsorbed molecules (11). In Fig. 1 C and D, one such tiling defect is identified. They are also entropically stabilized ''random tilings'' displaying algebraic spatial correlations (11), characteristic of a critical, or Coulomb,8). The structures, such as those in Fig. 1, are close to dynamically arrested at room temperature (11). The interaction energy between two neighboring molecules is several times k B T (11), so once a tiling is formed, tile removal is highly suppressed and structural relaxation is slow. Tile rearrangements mediated by propagation of defects have been observed experimentally, but so far on timescales of seconds for each event (11). This combination of an amorphous structure, albeit with critical spatial correlat...

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Classical dimers with aligning interactions on the square lattice

Cited by 105 publications

References 58 publications

Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets

Topological Aspects of Symmetry Breaking in Triangular-Lattice Ising Antiferromagnets

Ising nematic fluid phase of hard-core dimers on the square lattice

Molecular random tilings as glasses

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