Statistical Mechanics of Dimers on a Plane Lattice. II. Dimer Correlations and Monomers

Fisher, Michael E.; Stephenson, John

doi:10.1103/physrev.132.1411

Cited by 263 publications

(284 citation statements)

References 25 publications

(15 reference statements)

Supporting

Mentioning

269

Contrasting

Unclassified

Order By: Relevance

“…In strictly two dimensions the monomer-monomer cor- relations for N = 1 were calculated in [11,25]. It was found that η ∼ 0.5, which was recently confirmed again in [6].…”

mentioning

confidence: 81%

“…They have been studied extensively in the context of statistical and condensed matter physics. In the sixties, these models attracted a lot of attention [10,11] when it was shown that the Ising model can be rewritten as a dimer model [12]. In the late eighties, these models became fashionable again in the quantum version [13] as a promising approach to the famous Resonating-ValenceBond (RVB) liquid phase [14].…”

mentioning

confidence: 99%

See 1 more Smart Citation

Kosterlitz-Thouless universality in dimer models

Chandrasekharan

Strouthos

2003

Phys. Rev. D

View full text Add to dashboard Cite

Using the monomer-dimer representation of strongly coupled U (N ) lattice gauge theories with staggered fermions, we study finite temperature chiral phase transitions in (2 + 1) dimensions. A new cluster algorithm allows us to compute monomer-monomer and dimer-dimer correlations at zero monomer density (chiral limit) accurately on large lattices. This makes it possible to show convincingly, for the first time, that these models undergo a finite temperature phase transition which belongs to the Kosterlitz-Thouless universality class. We find that this universality class is unaffected even in the large N limit. This shows that the mean field analysis often used in this limit breaks down in the critical region.Computing quantities in Lattice QCD with massless quarks is notoriously difficult. Most known algorithms break down in the chiral limit. For this reason questions related to the universality of chiral phase transitions are among the many questions that remain unanswered. It is often difficult to compute critical exponents sufficiently accurately to rule out all possibilities except one.The most useful simplification of lattice QCD occurs in the strong coupling limit which retains much of the underlying physics of QCD except for large lattice artifacts. In this limit spontaneous chiral symmetry breaking and its restoration due to finite temperature effects have been studied using large N and large d expansions [1][2][3]. However, since these approaches are based on mean field analysis they cannot help in determining the universality of phase transitions.Interestingly lattice QCD with staggered fermions interacting through U (N ) gauge fields can be mapped into a monomer-dimer system in the strong coupling limit [4]. These models contain an exact U (1) chiral symmetry, a remnant of the full chiral symmetry of QCD. When it was proposed, the monomer-dimer representation offered a new approach to study strongly coupled gauge theories close to the chiral limit from first principles. Unfortunately, this dream has remained unfulfilled until now. As in the weak coupling regime, most numerical simulations of the monomer-dimer systems have suffered from critical slowing down close to the chiral limit and hence have only allowed calculations with limited accuracy [5].Recently, a cluster algorithm has been discovered to study these strongly coupled lattice gauge theories in the chiral limit [6]. This allows precision calculations in the chiral limit for the first time. As a first application of this new algorithm, in this article we study the finite temperature critical behavior in (2 + 1) dimensions. In agreement with expectations from universality, we find with very high precision the chiral phase transition to be in the same universality class as the Berezinski-KosterlitzThouless (BKT) transition [7]. Our results are comparable to other known high precision spin-model studies of this universality class. We also show that the BKT transition persists even in the large N limit, showing that the mean field analysis breaks ...

show abstract

“…In strictly two dimensions the monomer-monomer cor- relations for N = 1 were calculated in [11,25]. It was found that η ∼ 0.5, which was recently confirmed again in [6].…”

mentioning

confidence: 81%

mentioning

confidence: 99%

Kosterlitz-Thouless universality in dimer models

Chandrasekharan

Strouthos

2003

Phys. Rev. D

View full text Add to dashboard Cite

show abstract

“…In the following, we consider specifically the -flux lattice model with free bc; the lattice propagator ͑Dirac fermions͒ can be found, e.g., in Ref. 25, but here we need only note that G 0 (r 1 ,r 2 ) is bounded and behaves as ͉G 0 (r 1 ,r 2 )͉ϳ͉r 1 Ϫr 2 ͉ Ϫ1 for large ͉r 1 Ϫr 2 ͉ӷ1. A simple ͑completely rigorous͒ bound ʈ t AB Ϫ1 ʈрconstϫZ(L) immediately follows from the boundedness of G 0 .…”

Section: A the Random Vector Potential Modelmentioning

confidence: 99%

Particle-hole symmetric localization in two dimensions

2002

View full text Add to dashboard Cite

We revisit two-dimensional particle-hole symmetric sublattice localization problem, focusing on the origin of the observed singularities in the density of states (E) at the band center Eϭ0. The most general system of this kind ͓R. Gade, Nucl. Phys. B 398, 499 ͑1993͔͒ exhibits critical behavior and has (E) that diverges stronger than any integrable power law, while the special random vector potential model of Ludwig et al. ͓Phys. Rev. B 50, 7526 ͑1994͔͒ has instead a power-law density of states with a continuously varying dynamical exponent. We show that the latter model undergoes a dynamical transition with increasing disorder-this transition is a counterpart of the static transition known to occur in this system; in the strong-disorder regime, we identify the low-energy states of this model with the local extrema of the defining two-dimensional Gaussian random surface. Furthermore, combining this ''surface fluctuation'' mechanism with a renormalization group treatment of a related vortex glass problem leads us to argue that the asymptotic low-E behavior of the density of states in the general case is (E)ϳE Ϫ1 e Ϫc͉ln E͉ 2/3 , different from earlier prediction of Gade. We also study the localized phases of such particle-hole symmetric systems and identify a Griffiths ''string'' mechanism that generates singular power-law contributions to the low-energy density of states in this case.

show abstract

“…10 For the square lattice with hardcore dimer covering, the correlation of a pair of monomers has been obtained. 11,12 The main findings of the present work are the following: For the square and the honeycomb lattice, correlations as a function of distance show a power-law behavior, with an exponent 1/2 for the hard-core dimer covering on the square lattice and 1/3 for a loop dimer covering. The honeycomb lattice has an exponent of 1/2 for both coverings.…”

Section: Introductionmentioning

confidence: 84%

Classical correlations of defects in lattices with geometrical frustration in the motion of a particle

2006

View full text Add to dashboard Cite

We map certain highly correlated electron systems on lattices with geometrical frustration in the motion of added particles or holes to the spatial defect-defect correlations of dimer models in different geometries. These models are studied analytically and numerically. We consider different coverings for four different lattices: square, honeycomb, triangular, and diamond. In the case of a hard-core dimer covering, we verify the existing results for square and triangular lattices and obtain new ones for the honeycomb and diamond lattices while in the case of a loop covering we obtain new numerical results for all the lattices and use the existing analytical Liouville field theory for the case of a square lattice. The results show power-law correlations for the square and honeycomb lattices, while exponential decay with distance is found for the triangular lattice and exponential decay with the inverse distance on the diamond lattice. We relate this fact to the lack of bipartiteness of the triangular lattice and in the latter case to the three dimensionality of the diamond. The connection of our findings to the problem of fractionalized charge in such lattices is pointed out.

show abstract

Statistical Mechanics of Dimers on a Plane Lattice. II. Dimer Correlations and Monomers

Cited by 263 publications

References 25 publications

Kosterlitz-Thouless universality in dimer models

Kosterlitz-Thouless universality in dimer models

Particle-hole symmetric localization in two dimensions

Classical correlations of defects in lattices with geometrical frustration in the motion of a particle

Contact Info

Product

Resources

About