Although the physical properties of the 2D and 1D Ising models are quite different, we point out an interesting connection between their complex-temperature phase diagrams. We carry out an exact determination of the complex-temperature phase diagram for the 1D Ising model for arbitrary spin s and show that in the u_s=e^{-K/s^2} plane (i) it consists of N_{c,1D}=4s^2 infinite regions separated by an equal number of boundary curves where the free energy is non-analytic; (ii) these curves extend from the origin to complex infinity, and in both limits are oriented along the angles \\theta_n = (1+2n)\\pi/(4s^2), for n=0,..., 4s^2-1; (iii) of these curves, there are N_{c,NE,1D}=N_{c,NW,1D}=[s^2] in the first and second (NE and NW) quadrants; and (iv) there is a boundary curve (line) along the negative real u_s axis if and only if s is half-integral. We note a close relation between these results and the number of arcs of zeros protruding into the FM phase in our recent calculation of partition function zeros for the 2D spin s Ising model.Although the physical properties of the 2D and 1D Ising models are quite different, we point out an interesting connection between their complex-temperature phase diagrams. We carry out an exact determination of the complex-temperature phase diagram for the 1D Ising model for arbitrary spin s and show that in the u_s=e^{-K/s^2} plane (i) it consists of N_{c,1D}=4s^2 infinite regions separated by an equal number of boundary curves where the free energy is non-analytic; (ii) these curves extend from the origin to complex infinity, and in both limits are oriented along the angles \\theta_n = (1+2n)\\pi/(4s^2), for n=0,..., 4s^2-1; (iii) of these curves, there are N_{c,NE,1D}=N_{c,NW,1D}=[s^2] in the first and second (NE and NW) quadrants; and (iv) there is a boundary curve (line) along the negative real u_s axis if and only if s is half-integral. We note a close relation between these results and the number of arcs of zeros protruding into the FM phase in our recent calculation of partition function zeros for the 2D spin s Ising model.

Jan 1, 199416 citations

Complex-temperature phase diagrams of one-dimensional spin models with next-nearest-neighbor couplings

Robert E. Shrock, Shan-Ho Tsai

We study the dependence of complex-temperature phase diagrams on details of the Hamiltonian, focusing on the effect of non-nearest-neighbor spin-spin couplings. For this purpose, we consider a simple exactly solvable model, the one-dimensional (1D) Ising model with nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings. We work out the exact phase diagrams for various values of ${\mathrm{J}}_{\mathrm{NNN}}$ / ${\mathrm{J}}_{\mathrm{NN}}$ and compare these with the case of pure NN couplings. We also give some similar results for the 1D Potts model with NN and NNN couplings.We study the dependence of complex-temperature phase diagrams on details of the Hamiltonian, focusing on the effect of non-nearest-neighbor spin-spin couplings. For this purpose, we consider a simple exactly solvable model, the one-dimensional (1D) Ising model with nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings. We work out the exact phase diagrams for various values of ${\mathrm{J}}_{\mathrm{NNN}}$/${\mathrm{J}}_{\mathrm{NN}}$ and compare these with the case of pure NN couplings. We also give some similar results for the 1D Potts model with NN and NNN couplings.

arXiv:cond-mat/9703187Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsMay 1, 19977 citations

Phase diagram in the generalized chiral clock models

Helen Au-Yang, Jacques H. H. Perk

Physica A Statistical Mechanics and its ApplicationsJun 1, 19963 citations

Monte Carlo study of the phase diagram for the two-dimensional Z(4) model

Gilson Carneiro, Maria Elena Pol, Nicim Zagury

Physics Letters ANov 1, 19825 citations

A complex short-range order phase diagram in a one-dimensional spin model

Andrzej Fleszar, J. Glazer, G. Baskaran

The authors study in detail the short-range order at low temperature in a one-dimensional Ising model with competing interactions by studying various staggered susceptibilities. From this they get a 'phase diagram' which divides regions of various dominant short-range orders. This phase diagram is shown to resemble qualitatively the low-temperature phase diagram of the anisotropic next-nearest-neighbour interaction Ising (ANNNI) model in higher dimensions. They argue that this method of studying a corresponding one-dimensional problem may be useful to understand the phase structure of complex higher-dimensional systems.The authors study in detail the short-range order at low temperature in a one-dimensional Ising model with competing interactions by studying various staggered susceptibilities. From this they get a 'phase diagram' which divides regions of various dominant short-range orders. This phase diagram is shown to resemble qualitatively the low-temperature phase diagram of the anisotropic next-nearest-neighbour interaction Ising (ANNNI) model in higher dimensions. They argue that this method of studying a corresponding one-dimensional problem may be useful to understand the phase structure of complex higher-dimensional systems.

Journal of Physics C Solid State PhysicsSep 30, 19852 citations

{\mathrm{J}}_{\mathrm{NNN}}

{\mathrm{J}}_{\mathrm{NN}}

and compare these with the case of pure NN couplings. We also give some similar results for the 1D Potts model with NN and NNN couplings.