The antiferromagnetic transition for the square-lattice Potts model

Abstract

We solve the antiferromagnetic transition for the Q-state Potts model (defined geometrically for Q generic) on the square lattice. The solution is based on a detailed analysis of the Bethe ansatz equations (which involve staggered source terms) and on extensive numerical diagonalization of transfer matrices. It involves subtle distinctions between the loop/cluster version of the model, and the associated RSOS and (twisted) vertex models. The latter's continuum limit involves two bosons, one which is compact and twisted, and the other which is not, with a total central charge c=2-6/t, for sqrt(Q)=2cos(pi/t). The non-compact boson contributes a continuum component to the spectrum of critical exponents. For Q generic, these properties are shared by the Potts model. For Q a Beraha number [Q = 4 cos^2(pi/n) with n integer] the two-boson theory is truncated and becomes essentially Z\_{n-2} parafermions. Moreover, the vertex model, and, for Q generic, the Potts model, exhibit a first-order critical point on the transition line, i.e., the critical point is also the locus of level crossings where the derivatives of the free energy are discontinuous. In that sense, the thermal exponent of the Potts model is generically nu=1/2. Things are profoundly different for Q a Beraha number, where the transition is second order, with nu=(t-2)/2 determined by the psi\_1 parafermion. As one enters the adjacant Berker-Kadanoff phase, the model flows, for t odd, to a minimal model of CFT with c=1-6/t(t-1), while for t even it becomes massive. This provides a physical realization of a flow conjectured by Fateev and Zamolodchikov in the context of Z\_N integrable perturbations. Finally, we argue that the antiferromagnetic transition occurs as well on other two-dimensional lattices.

References in corpus (8)

• Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions
• Dense loops, supersymmetry, and Goldstone phases in two dimensions
• Continuum Limit of the Integrable sl(2/1) 3-\bar{3} Superspin Chain
• Spanning forests and the q-state Potts model in the limit q \to 0
• Lattice regularization of massive and massless integrable field theories
• The arboreal gas and the supersphere sigma model
• The Exact S-Matrix for an osp(2|2) Disordered System
• Exact renormalization-group analysis of first order phase transitions in clock models

Cited by in corpus (60)

• Probing symmetries of quantum many-body systems through gap ratio statistics
• An integrable spin chain for the SL(2,R)/U(1) black hole sigma model
• Non compact conformal field theory and the a_2^{(2)} (Izergin-Korepin) model in regime III
• The Staggered Six-Vertex Model: Conformal Invariance and Corrections to Scaling
• Phase Diagram of an Integrable Alternating $U_q[sl(2|1)]$ Superspin Chain
• The $Z_2$ staggered vertex model and its applications
• Critical manifold of the kagome-lattice Potts model
• Eigenvalue amplitudes of the Potts model on a torus
• Scaling limit of the ${\mathcal{Z}}_2$ invariant inhomogeneous six-vertex model
• Non-Linear Integral Equations for the SL(2,R)/U(1) black hole sigma model
• Transfer matrix computation of critical polynomials for two-dimensional Potts models
• Corner free energies and boundary effects for Ising, Potts and fully-packed loop models on the square and triangular lattices
• Odd q-State Clock Spin-Glass Models in Three Dimensions, Asymmetric Phase Diagrams, and Multiple Algebraically Ordered Phases
• Classifying Potts critical lines
• Tensor network approach to phase transitions of a non-Abelian topological phase
• Critical points in coupled Potts models and critical phases in coupled loop models
• The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic $Q$
• Complex-temperature phase diagram of Potts and RSOS models
• Integrable boundary conditions in the antiferromagnetic Potts model
• On the scaling behaviour of the alternating spin chain
• Lattice regularisation of a non-compact boundary conformal field theory
• Devil's Staircase Continuum in the Chiral Clock Spin Glass with Competing Ferromagnetic-Antiferromagnetic and Left-Right Chiral Interactions
• Reclaiming the Lost Conformality in a non-Hermitian Quantum 5-state Potts Model
• The Hintermann-Merlini-Baxter-Wu and the Infinite-Coupling-Limit Ashkin-Teller Models
• Renormalization flow for unrooted forests on a triangular lattice
• Q-colourings of the triangular lattice: Exact exponents and conformal field theory
• Conformally invariant boundary conditions in the antiferromagnetic Potts model and the $SL(2,\mathbb{R})/U(1)$ sigma model
• The continuum limit of $a_{N-1}^{(2)}$ spin chains
• Is the five-flow conjecture almost false?
• The antiferromagnetic Potts model
• Bethe state norms for the Heisenberg spin chain in the scaling limit
• Particles, conformal invariance and criticality in pure and disordered systems
• Scattering and duality in the 2 dimensional OSP(2|2) Gross Neveu and sigma models
• Some Algebraic Aspects of the Inhomogeneous Six-Vertex Model
• Phase diagram of the triangular-lattice Potts antiferromagnet
• A generalized Beraha conjecture for non-planar graphs
• An Ising-type formulation of the six-vertex model
• Character decomposition of Potts model partition functions. II. Toroidal geometry
• Entanglement entropy and massless phase in the antiferromagnetic three-state quantum chiral clock model
• Equilibrium density matrices for the 2D black hole sigma models from an integrable spin chain
• Elaborating the phase diagram of spin-1 anyonic chains
• On the scaling behaviour of an integrable spin chain with ${\mathcal{Z}}_r$ symmetry
• Non compact continuum limit of two coupled Potts models
• Integrability in anyonic quantum spin chains via a composite height model
• Spectrum of the quantum integrable $D_2^{(2)}$ spin chain with generic boundary fields
• Critical points in coupled Potts models and correlated percolation
• Critical Behaviour of Spanning Forests on Random Planar Graphs
• Exact surface energy of the $D_2^{(1)}$ spin chain with generic non-diagonal boundary reflections
• Original electric-vertex formulation of the symmetric eight-vertex model on the square lattice is fully non-universal
• Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice
• Potts critical frontiers of inhomogeneous and asymmetric bow-tie lattices
• Scaling limit of the staggered six-vertex model with $U_q(\mathfrak{sl}(2))$ invariant boundary conditions
• Spin chains with boundary inhomogeneities
• Loop models for CFTs
• Managing Singular Kernels and Logarithmic Corrections in the Staggered Six-Vertex Model
• Critical manifold of the Potts model: Exact results and homogeneity approximation
• Scaling limit of the ground state Bethe roots for the inhomogeneous XXZ spin-$\frac{1}{2}$ chain
• Critical Percolation Phase and Thermal BKT Transition in a Scale-Free Network with Short-Range and Long-Range Random Bonds
• Finite-size spectrum of the staggered six-vertex model with antidiagonal boundary conditions
• Factorization identities and algebraic Bethe ansatz for $D_2^{(2)}$ models