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Journal of Physics C Solid State Physics

Self-dual quantum Potts model with multispin interactions in one dimension

Loïc Turban, Jean‐Marc Debierre

Published Feb 28, 1982• 9 citations
Publisher

Abstract

The quantum Hamiltonian formulation of the q-state Potts model on a linear chain is generalised by introducing models with m-spin interactions. These models are self-dual for any values of q and m and have a first-order transition when q and/or m are large enough. The line in the (q,m) plane where the latent heat vanishes is approximately located using 1/q and 1/m expansions.

DOI
10.1088/0022-3719/15/6/002

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The quantum Hamiltonian formulation of the q-state Potts model on a linear chain is generalised by introducing models with m-spin interactions. These models are self-dual for any values of q and m and have a first-order transition when q and/or m are large enough. The line in the (q,m) plane where the latent heat vanishes is approximately located using 1/q and 1/m expansions.
Introduces a family of D-dimensional q-state quantum Potts models with multi-spin interactions, which are self-dual for any D. These models are introduced using their (d=D+1)-dimensional classical formulations and the transfer matrix technique. The self-duality is proved on the quantum Hamiltonian. In the limit q to infinity the ground state energy is obtained exactly and the phase transition is first-order. A 1/q expansion allows one to get an approximate expression for the line in the (q,D)-plane separating regions where the transition changes from first- to second-order.
The two-dimensional Ising model in the rectangular lattice is generalised to include m-spin interactions in one direction and two-spin interactions in the other. This model is self-dual and the critical line is the same as in the conventional Ising model with m=2. The m=3 model is solved in a generalised mean-field approximation where the transition is first order. Using a phenomenological renormalisation group approach, approximate values of the thermal and magnetic exponents y t and y h are obtained for m=3 and m=4. The m=3 results together with symmetry arguments suggest that this model belongs to the same class of universality as the Baxter-Wu and four-state ferromagnetic Potts models. When m=4 the transition is probably first order.
Abstract A one-dimensional (1D) q -state Potts model with N sites, m -site interaction K in a field H is studied for arbitrary values of m . Exact results for the partition function and the two-point correlation function are obtained at H = 0. The system in a field is shown to be self-dual. Using a change of Potts variables, it is mapped onto a standard 2D Potts model, with first-neighbour interactions K and H , on a cylinder with helical boundary conditions (BC). The 2D system has a length N / m and a transverse size m . Thus the Potts chain with multi-site interactions is expected to develop a 2D critical singularity along the self-duality line, ( e q K − 1 ) ( e q H − 1 ) = q , when N / m → ∞ and m → ∞ .
Generalizing the mapping between the Potts model with nearest neighbor interaction and the six-vertex model, we build a family of “fused Potts models” related to the spin k/2 U q su (2) invariant vertex model and quantum spin chain. These Potts models still have variables taking values 1, …,Q[Formula: see text] but they have a set of complicated multispin interactions. The general technique to compute these interactions, the resulting lattice geometry, symmetries, and the detailed examples of k=2, 3 are given. For Q>4, spontaneous magnetizations are computed on the integrable first order phase transition line, generalizing Baxter’s results for k=1. For Q≤4, we discuss the full phase diagram of the spin 1 (k=2) anisotropic and U q su (2) invariant quantum spin chain [it reduces in the limit Q=4 (q=1) to the much studied phase diagram of the isotropic spin 1 quantum spin chain], Several critical lines and massless phases are exhibited. The appropriate generalization of the valence bond state method of Affleck et al. is worked out.
The one-dimensional Hamiltonian version of the two-dimensional Potts model is studied by the application of a self-dual renormalization-group transformation. Blocks containing up to 11 sites are considered and mapped onto new, equivalent Potts spins by keeping the lowest-lying levels only. The higher-lying states are also taken into account in a perturbational way for smaller cells. Unfortunately, this renormalization-group treatment cannot reproduce the first-order transition for large values of the number of states, and the critical exponents for the three- and four-state Potts model do not seem to converge to the conjectured results as larger and larger blocks are taken.