Yang-Baxter Integrable Dimers on a Strip

Abstract

The dimer model on a strip is considered as a Yang-Baxter \mbox{integrable} six vertex model at the free-fermion point with crossing parameter $\lambda =\frac{\pi }{2}$ and quantum group invariant boundary conditions. A one-to-many mapping of vertex onto dimer configurations allows for the solution of the free-fermion model to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by $45°$ compared to their usual orientation. In a suitable gauge, the dimer model is described by the Temperley-Lieb algebra with loop fugacity $\beta =2\cos \lambda =0$. It follows that the model is exactly solvable in geometries of arbitrary finite size. We establish and solve transfer matrix inversion identities on the strip with arbitrary finite width $N$. In the continuum scaling limit, in sectors with magnetization $S_z$, we obtain the conformal weights ${\Delta }_s=((2-s{)}^2-1)/8$ where $s=|S_z|+1=1,2,3,\dots$. We further show that the corresponding finitized characters $\chit_s^{(N)}(q)$ decompose into sums of $q$-Narayana numbers or, equivalently, skew $q$-binomials. In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal degrees of freedom. We argue that, in the continuum scaling limit, there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation operator $L_0$. This confirms that, with quantum group invariant boundary conditions, the dimer model gives rise to a {\em logarithmic} conformal field theory with central charge $c=-2$, minimal conformal weight ${\Delta }_{\text{min}}=-\frac{1}{8}$ and effective central charge $c_{\text{eff}}=1$.Our analysis of the structure of the ensuing rank 2 modules indicates that the familiar staggered $c=-2$ modules appear as submodules.

arXiv comment35 pages, 10 Figures. arXiv admin note: text overlap with arXiv:1612.09477

References in corpus (26)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• A Rational Logarithmic Conformal Field Theory
• Symplectic Fermions
• Polymers and percolation in two dimensions and twisted N=2 supersymmetry
• A Local Logarithmic Conformal Field Theory
• Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions
• Interaction-Round-a-Face Models with Fixed Boundary Conditions: The ABF Fusion Hierarchy
• Solvable Critical Dense Polymers
• Logarithmic Conformal Field Theory and Boundary Effects in the Dimer Model
• From boundary to bulk in logarithmic CFT
• Domino tilings and the six-vertex model at its free fermion point
• Solvable Critical Dense Polymers on the Cylinder
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• Modular invariant partition function of critical dense polymers
• Lattice W-algebras and logarithmic CFTs
• Exact solution of the $2d$ dimer model: Corner free energy, correlation functions and combinatorics
• Integrability and conformal data of the dimer model
• A Proof of Selection Rules for Critical Dense Polymers
• Non-Local Finite-Size Effects in the Dimer Model
• Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
• Finite-Volume Spectra of the Lee-Yang Model
• On the reality of spectra of ${\mathbf{U}}_{\mathbf{q}}(\mathbf{s}{\mathbf{l}}_2)$-invariant XXZ Hamiltonians
• Dimer representations of the Temperley-Lieb algebra
• Refined conformal spectra in the dimer model
• RSOS Quantum Chains Associated with Off-Critical Minimal Models and ${\mathbb{Z}}_n$ Parafermions
• Yang-Baxter Solution of Dimers as a Free-Fermion Six-Vertex Model

Cited by in corpus (3)

• Spin-Ruijsenaars, q-deformed Haldane-Shastry and Macdonald polynomials
• Critical site percolation on the triangular lattice: From integrability to conformal partition functions
• Integrability of planar-algebraic models