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Journal of Physics A Mathematical and General

Self-dual D-dimensional quantum Potts model with multi-spin interactions

Loïc Turban

Published May 1, 1982• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/15/5/006

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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.
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 → ∞ .
We establish explicit duality transformations for systems of M q-state Potts models coupled through their local energy density, generalizing known results for M=1,2,3. The M-dimensional space of coupling constants contains a self-dual submanifold of dimension D(M)=[M/2]. For the case M=4, the variation of the effective central charge along the self-dual surface is investigated by numerical transfer matrix techniques. Evidence is given for the existence of a family of critical points, corresponding to conformal field theories with an extended S(M) symmetry algebra.
A quantum model with nnn-spin interactions is studied in the ground state. When $n\ensuremath{\rightarrow}\ensuremath{\infty}$ the model is solved exactly by mean-field theory and undergoes a first-order transition. The transition is expected to become second order for n<ncn<{n}_{c}n<nc​. With exploitation of self-duality and suitably improved mean-field approximations, a recently proposed criterion leads to the conjecture nc=3{n}_{c}=3nc​=3 in one dimension. For the two-dimensional case the transition appears to be definitely of first order for n=3(2<nc<3)n=3 (2<{n}_{c}<3)n=3(2<nc​<3).
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.
Abstract Five duality transformations are unveiled for the quantum XYZ model with arbitrary spin s in one spatial dimension. The presence of these duality transformations, together with an extra Hamiltonian symmetry, drastically reduce the entire ground-state phase diagram to two finite regimes—the principal regimes, with all the other ten regimes dual to them. Combining with the determination of critical points from the conventional order parameter approach and/or the fidelity approach to quantum phase transitions, we are able to map out the ground-state phase diagram for the quantum XYZ model with arbitrary spin s . This is explicitly demonstrated for s = 1 / 2 , 1 , 3 / 2 and 2. As it turns out, all the critical points, with central charge c = 1 , are self-dual for half-integer as well as integer spin s . However, in the latter case, the presence of the Haldane phase results in extra lines of critical points with central charge c = 1 / 2 , which are not self-dual.