scholarly journals The Toggle Group, Homomesy, and the Razumov-Stroganov Correspondence

10.37236/5158 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Jessica Striker
Keyword(s):  
Ground State ◽  
Statistical Physics ◽  
Infinite Cylinder ◽  
Loop Model ◽  
Alternating Sign Matrices ◽  
Important Link

The Razumov-Stroganov correspondence, an important link between statistical physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello, relates the ground state eigenvector of the $O(1)$ dense loop model on a semi-infinite cylinder to a refined enumeration of fully-packed loops, which are in bijection with alternating sign matrices. This paper reformulates a key component of this proof in terms of posets, the toggle group, and homomesy, and proves two new homomesy results on general posets which we hope will have broader implications.

scholarly journals Proof of the Razumov–Stroganov Conjecture for some Infinite Families of Link Patterns

10.37236/1136 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
P. Zinn-Justin
Keyword(s):  
Ground State ◽  
Loop Model ◽  
Plane Partitions

We prove the Razumov–Stroganov conjecture relating ground state of the $O(1)$ loop model and counting of Fully Packed Loops in the case of certain types of link patterns. The main focus is on link patterns with three series of nested arches, for which we use as key ingredient of the proof a generalization of the MacMahon formula for the number of plane partitions which includes three series of parameters.

scholarly journals Logarithmic correlation functions for critical dense polymers on the cylinder

2019 ◽  
Vol 7 (3) ◽  
Author(s):  
Alexi Morin-Duchesne ◽  
Jesper Jacobsen
Keyword(s):  
Boundary Condition ◽  
Correlation Functions ◽  
Central Charge ◽  
Infinite Cylinder ◽  
Operator Product ◽  
Partition Functions ◽  
Loop Model ◽  
Perfect Agreement ◽  
Single Node ◽  
Highest Weight

We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter nn. In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity \alpha \in (0,\infty)α∈(0,∞). These correlators are defined as ratios Z(x)/Z_0Z(x)/Z0 of partition functions, where Z_0Z0 is a reference partition function wherein only simple half-arcs are attached to the boundary of the cylinder. For Z(x)Z(x), the boundary of the cylinder is also decorated with simple half-arcs, but it also has two special positions 11 and xx where the boundary condition is different. We investigate two such kinds of boundary conditions: (i) there is a single node at each of these points where a long arc is attached, and (ii) there are pairs of adjacent nodes at these points where two long arcs are attached. We find explicit expressions for these correlators for finite nn using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as n\to \inftyn→∞ are expressed as simple integrals that depend on the scaling parameter \tau = \frac {x-1} n \in (0,1)τ=x−1n∈(0,1). For small \tauτ, the leading behaviours are proportional to \tau^{1/4}τ1/4, \tau^{1/4}\log \tauτ1/4logτ, \log \taulogτ and \log^2 \taulog2τ. We interpret the lattice results in terms of ratios of conformal correlation functions. We assume that the corresponding boundary changing fields are highest weight states in irreducible, Kac or staggered Virasoro modules, with central charge c=-2c=−2 and conformal dimensions \Delta = -\frac18Δ=−18 or \Delta = 0Δ=0. With these assumptions, we obtain differential equations of order two and three satisfied by the conformal correlation functions, solve these equations in terms of hypergeometric functions, and find a perfect agreement with the lattice results. We use the lattice results to compute structure constants and ratios thereof which appear in the operator product expansions of the boundary condition changing fields. The fusion of these fields is found to be non-abelian.

Selective Quantum Annealing Using Transverse XY-Type Interaction

2012 ◽  
Vol 452-453 ◽  
pp. 1460-1464 ◽  
Author(s):  
Yohei Saika ◽  
Tetsuya Kakimoto ◽  
Jun Ichi Inoue
Keyword(s):  
Ground State ◽  
Statistical Physics ◽  
Wave Theory ◽  
Kinetic Term ◽  
Target System ◽  
Quantum Annealing ◽  
Initial State ◽  
Type Interaction ◽  
Quantum Spins ◽  
Ferromagnetic Interactions

We investigated quantum annealing (QA) via the transverse interaction with XY-type anisotropy for a ground state problem for a small composed of 4 S=1/2 quantum spins interacting with anti-ferromagnetic interactions with each other. By solving the Schrodinger equation for the QA system, we found that a preferable solution can be derived by tuning the XY-type anisotropy of the kinetic term among multiple candidates of the QA system. Similar behavior was suggested from the static property obtained by the spin wave theory established in statistical physics. In addition, we clarified that the ground state of the target system can be obtained by the QA starting from an initial state including excited states of the kinetic term, if the interval of time of the QA is set to be large to some extent.

scholarly journals Exact expressions for correlations in the ground state of the dense O(1) loop model

2004 ◽  
Vol 2004 (09) ◽  
pp. P09010 ◽  
Author(s):  
S Mitra ◽  
B Nienhuis ◽  
J de Gier ◽  
M T Batchelor
Keyword(s):  
Ground State ◽  
Loop Model

scholarly journals Around the Razumov–Stroganov Conjecture: Proof of a Multi-Parameter Sum Rule

10.37236/1903 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
P. Di Francesco ◽  
P. Zinn-Justin
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Loop Model ◽  
Vertex Model ◽  
Sum Rule ◽  
Alternating Sign Matrices ◽  
Wall Boundary Conditions ◽  
Periodic Strip ◽  
Domain Wall Boundary Conditions

We prove that the sum of entries of the suitably normalized groundstate vector of the $O(1)$ loop model with periodic boundary conditions on a periodic strip of size $2n$ is equal to the total number of $n\times n$ alternating sign matrices. This is done by identifying the state sum of a multi-parameter inhomogeneous version of the $O(1)$ model with the partition function of the inhomogeneous six-vertex model on a $n\times n$ square grid with domain wall boundary conditions.

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