The Kosterlitz-Thouless phase transition in two-dimensional hierarchical Coulomb gases

Marchetti, Domingos H. U.; Perez, J. Fernando

doi:10.1007/bf01042594

Cited by 14 publications

(15 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…I am only going to discuss the dipole phase of the hierarchical model. It is an artifact of this model that the transition from confinement to screening phase is much less noticeable than in the Euclidean case; according to [43], all that happens is that whereas the dielectric constant ε = 1 in the dipole phase, it is less than one in the screening phase. Furthermore, the hierarchical model has a simpler dipole phase than the Euclidean model, as manifested by ε = 1, meaning that E a,b is behaving as if all the other particles were not there; they are bound into neutral pairs in a stronger sense than in the Euclidean model, for which ε = 1 in the dipole phase.…”

Section: Example: Hierarchical Coulomb Gasmentioning

confidence: 99%

“…For an illustration we turn to the Coulomb gas on the hierarchical lattice. We are following Marchetti and Perez in [43], but it looks more elaborate here because we are preparing the way for models on the Euclidean lattice as well. The goal is to evaluate the asymptotics as a − b → ∞ for the energy of a pair of fractional charges at positions a, b in order to see if they are "confined".…”

Section: Example: Hierarchical Coulomb Gasmentioning

confidence: 99%

“…In [43], the factor e λ0 hardly appears because it cancels out in correlations, and it is usual to take advantage of this fact to get rid of it, but I prefer to let it remain as it is the very simplest example of a general phenomenon in RG. λ 0 is a coordinate for the expanding direction under the action of (RG) in the spaces N Bj j .…”

Section: )mentioning

confidence: 99%

See 2 more Smart Citations

Lectures on the renormalisation group

Brydges¹

2009

Statistical Mechanics

116

View full text Add to dashboard Cite

show abstract

Section: Example: Hierarchical Coulomb Gasmentioning

confidence: 99%

Section: Example: Hierarchical Coulomb Gasmentioning

confidence: 99%

See 1 more Smart Citation

Lectures on the renormalisation group

Brydges¹

2009

Statistical Mechanics

116

View full text Add to dashboard Cite

show abstract

“…to denote the obvious corresponding objects in D. Hierarchical models were introduced by Dyson in statistical mechanics to study a variant of the one-dimensional Ising model [22,23]. Subsequently, the so called Dyson's hierarchical model has led to a lot of activity, see for e.g., [7,44,21,40,8,32] for results about the two-dimensional hierarchical Coulomb gas. However, before [18] not much was known about this model in any dimension greater than 2.…”

Section: 1mentioning

confidence: 99%

Ground states and hyperuniformity of the hierarchical Coulomb gas in all dimensions

Ganguly

Sarkar

2019

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

Stochastic point processes with Coulomb interactions arise in various natural examples of statistical mechanics, random matrices and optimization problems. Often such systems due to their natural repulsion exhibit remarkable hyperuniformity properties, that is, the number of points landing in any given region fluctuates at a much smaller scale compared to that of a set of i.i.d. random points. A well known conjecture from physics appearing in the works of Jancovici, Lebowitz, Manificat, Martin, and Yalcin (see [43,46,37]), states that the variance of the number of points landing in a set should grow like the surface area instead of the volume unlike i.i.d. random points. In a recent beautiful work [18], Chatterjee gave the first proof of such a result in dimension three for a Coulomb type system, known as the hierarchical Coulomb gas, inspired by Dyson's hierarchical model of the Ising ferromagnet [22,23]. However the case of dimensions greater than three had remained open. In this paper, we establish the correct fluctuation behavior up to logarithmic factors in all dimensions greater than three, for the hierarchical model. Using similar methods, we also prove sharp variance bounds for smooth linear statistics which were unknown in any dimension bigger than two. A key intermediate step is to obtain precise results about the ground states of such models whose behavior can be interpreted as hierarchical analogues of various crystalline conjectures predicted for energy minimizing systems, and could be of independent interest.

show abstract

“…All these results are valid also in the hierarchical approximation of the model (see Sect. 2), where they are obtained much easier [5,6].…”

Section: Introductionmentioning

confidence: 99%

Nontrivial fixed points and screening in the hierarchical two-dimensional Coulomb gas

Benfatto¹,

Renn

1992

J Stat Phys

View full text Add to dashboard Cite

We show the existence and asymptotic stability of two fixed points of the renormalization group transformation for the hierarchical twodimensional Coulomb gas in the sine-Gordon representation and temperatures slightly greater than the critical one. We prove also that the correlations at the fixed points decay as in the hierarchical massive scalar free field theory, that is as d −4 xy. We argue that this is the natural definition of screening in the hierarchical approximation.

show abstract

The Kosterlitz-Thouless phase transition in two-dimensional hierarchical Coulomb gases

Cited by 14 publications

References 9 publications

Lectures on the renormalisation group

Lectures on the renormalisation group

Ground states and hyperuniformity of the hierarchical Coulomb gas in all dimensions

Nontrivial fixed points and screening in the hierarchical two-dimensional Coulomb gas

Contact Info

Product

Resources

About