KBD algorithm and Draft:KBD algorithm: Difference between pages

{{Short description|Cluster update algorithm}}

{{refimprove|date=May 2021}}

The '''KBD algorithm''' is a [[Swendsen–Wang algorithm|cluster update algorithm]] designed for the [[Domino tiling|fully frustrated Ising model]] in two dimensions,<ref name=":0">{{Cite journal|last1=Kandel|first1=Daniel|last2=Ben-Av|first2=Radel|last3=Domany|first3=Eytan|date=1990-08-20|title=Cluster dynamics for fully frustrated systems|url=https://link.aps.org/doi/10.1103/PhysRevLett.65.941|journal=Physical Review Letters|volume=65|issue=8|pages=941–944|doi=10.1103/PhysRevLett.65.941|pmid=10043065|bibcode=1990PhRvL..65..941K|url-access=subscription}}</ref> or more generally any two dimensional spin glass with frustrated plaquettes arranged in a [[Check (pattern)|checkered pattern]].<ref>{{Cite journal|last1=Hamze|first1=Firas|last2=Jacob|first2=Darryl C.|last3=Ochoa|first3=Andrew J.|last4=Perera|first4=Dilina|last5=Wang|first5=Wenlong|last6=Katzgraber|first6=Helmut G.|date=2018-04-13|title=From near to eternity: Spin-glass planting, tiling puzzles, and constraint-satisfaction problems|journal=Physical Review E|volume=97|issue=4|pages=043303|doi=10.1103/PhysRevE.97.043303|pmid=29758754|arxiv=1711.04083|bibcode=2018PhRvE..97d3303H|doi-access=free}}</ref> It is discovered in 1990 by Daniel Kandel, Radel Ben-Av, and Eytan Domany, and generalized by P. D. Coddington and L. Han in 1994.<ref name=coddington>{{Cite journal|last1=Coddington|first1=P. D.|last2=Han|first2=L.|date=1994-08-01|title=Generalized cluster algorithms for frustrated spin models|url=https://link.aps.org/doi/10.1103/PhysRevB.50.3058|journal=Physical Review B|volume=50|issue=5|pages=3058–3067|doi=10.1103/PhysRevB.50.3058|pmid=9976551|arxiv=cond-mat/9402030|bibcode=1994PhRvB..50.3058C|s2cid=939709}}</ref> It is the inspiration for cluster algorithms used in [[Quantum Monte Carlo|quantum monte carlo]] simulations.

== Motivation ==

The [[Swendsen–Wang algorithm|SW algorithm]] is the first non-local algorithm designed for efficient simulation of [[Ising model|ferromagnetic spin models]].<ref>{{Cite journal|last1=Swendsen|first1=Robert H.|last2=Wang|first2=Jian-Sheng|date=1987-01-12|title=Nonuniversal critical dynamics in Monte Carlo simulations|url=https://link.aps.org/doi/10.1103/PhysRevLett.58.86|journal=Physical Review Letters|volume=58|issue=2|pages=86–88|doi=10.1103/PhysRevLett.58.86|pmid=10034599|bibcode=1987PhRvL..58...86S|url-access=subscription}}</ref> However, it is soon realized that the efficiency of the algorithm cannot be extended to [[Geometrical frustration|frustrated systems]], due to an overly large [[Percolation critical exponents|correlation length of the generated clusters]] with respect to the underlying spin system.<ref>{{Cite journal|last=Cataudella|first=V.|date=1992-05-01|title=Percolation transition in systems with frustation|url=https://dx.doi.org/10.1016/0378-4371%2892%2990145-G|journal=Physica A: Statistical Mechanics and Its Applications|language=en|volume=183|issue=3|pages=249–254|doi=10.1016/0378-4371(92)90145-G|bibcode=1992PhyA..183..249C|issn=0378-4371|url-access=subscription}}</ref> The KBD algorithm is an attempt to extend the bond-formation rule to the plaquettes of the lattice, such that the generated clusters are informed by the frustration profile, resulting in them being smaller than the SW ones,<ref name=coddington/> thereby making the algorithm more efficient in comparison. However, at the current stage, it is not known whether this algorithm can be generalized for arbitrary [[spin glass]] models.

== Algorithm ==

We begin by decomposing the square lattice down into plaquettes arranged in a checkered pattern (such that the plaquettes only overlap vertex-wise but not edge-wise). Since the spin model is fully-frustrated, each plaquette must contain exactly one or three negative interactions.<ref name=":0" /> If the plaquette contains three negative interactions, then no bonds can be formed. However, if the plaquette contains one negative interaction, then two parallel bonds can be formed (perpendicular to the negative edge) with probability <math>p = 1-e^{-4\beta}</math>, where <math>\beta</math> is the [[Boltzmann distribution|inverse temperature]] of the spin model.

The bonds will then form [[Random cluster model|clusters]] on the lattice, on which the spins can be collectively flipped (either with the [[Swendsen–Wang algorithm|SW]] rule or the [[Wolff algorithm|Wolff]] rule ). It can be shown that the update satisfies [[detailed balance]], meaning that correctness is guaranteed if the algorithm is used in conjunction with [[Ergodicity|ergodic]] algorithms like [[Glauber dynamics|single spin-flip updates]].

== Topological features ==

At zero temperature, or the <math>\beta \to \infty</math> limit, all the plaquettes will contain exactly one negative edge. In this case, on each checkered plaquette, the KBD algorithm will always [[Percolation theory|open]] two parallel bonds perpendicular to the negative edge, meaning that the bond will be closed on the negative edge along with the edge opposite to it. If we were to track the closed bonds in the [[dual lattice]], by drawing a straight/bent line inside each plaquette such that it intersects with the closed bonds, then it can be shown that a path following the lines must form a [[Cycle (graph theory)|cycle]].

Furthermore, it can be shown that there must be at least two such cycles, and that the cycles cannot intersect. Most importantly, each cycle cannot be [[Homotopy|contracted]] to a point in the [[Euler characteristic|underlying surface]] that the lattice is [[Graph embedding|embedded]] in.<ref>{{Cite journal|last1=Kandel|first1=Daniel|last2=Ben-Av|first2=Radel|last3=Domany|first3=Eytan|date=1992-03-01|title=Cluster Monte Carlo dynamics for the fully frustrated Ising model|url=https://link.aps.org/doi/10.1103/PhysRevB.45.4700|journal=Physical Review B|volume=45|issue=9|pages=4700–4709|doi=10.1103/PhysRevB.45.4700|pmid=10002105|bibcode=1992PhRvB..45.4700K|url-access=subscription}}</ref> On a periodic lattice (or a [[torus]]), this means that the cycles of closed bonds must wind around the torus in the same direction, from which one can show that the [[Giant component|largest cluster]] (which must be "squeezed" between these cycles) at zero temperature cannot span a finite fraction of the lattice size in the [[thermodynamic limit]].

== References ==

{{reflist}}

{{Statistical mechanics topics}}

[[Category:Statistical mechanics]]

[[Category:Monte Carlo methods]]