Complex - Temperature Phase Diagram of the 1 D Z 6 Clock Model and its Connection with Higher - Dimensional Models

We determine the exact complex-temperature (CT) phase diagram of the 1D Z6 clock model. This is of interest because it is the first exactly solved system with a CT phase boundary exhibiting a finite-K intersection point where an odd number of curves (namely, three) meet, and yields a deeper insight into this phenomenon. Such intersection points occur in the 3D spin 1/2 Ising model and appear to occur in the 2D spin 1 Ising model. Further, extending our earlier work on the higher-spin Ising model, we point out an intriguing connection between the CT phase diagrams for the 1D and 2D Z6 clock models. ∗email: [email protected] ∗∗email: [email protected] We report here a determination of the exact complex-temperature phase diagram for the 1D Z6 clock model and show how the results give a deeper insight into certain features of higher-dimensional models. The idea of generalizing a variable, on which the free energy depends, from real physical values to complex values was pioneered by Yang and Lee [1], who carried this out for the external magnetic field and proved a celebrated circle theorem on the complex-field zeros of the Ising model partition function. The generalization of temperature to complex values, performed first for the Ising model [2], has also been quite fruitful, since it enables one to understand better the behavior of various thermodynamic quantities as analytic functions of complex temperature (CT) and to see how physical phases of a given model generalize to regions in CT variables. Indeed, a knowledge of the complex-temperature phase diagram of a model for which the free energy is not known provides further constraints to guide progress toward an exact solution. A classic example relevant here is the 3D Ising model; the main purpose of the present work is to present some exact results for a 1D model which elucidate an important feature of the complex-temperature phase diagram of the 3D Ising model. Some early works on CT singularities include Refs. [3]-[6] studying partition function zeros, and Ref. [7], motivated by the effect of these singularities on low-temperature series. The continuous locus of points in the complex-temperature plane where the free energy is non-analytic is denoted B. For the spin models of interest here, with isotropic, nearest-neighbor spin-spin couplings, this is a 1-dimensional curve (including possible line segments). Parts of B form boundaries of various regions, some of which are complex-temperature extensions of physical phases and some of which may have no overlap with any physical phase. B may also include arcs or line segments which protrude into, and terminate in, certain CT phases. In general, phase boundaries of physical systems include intersection points where different parts of the boundaries meet. Examples include (i) the triple point in the (T, p) phase diagram for a substance like argon, water, etc. where gas, liquid, and solid phases all coexist, and (ii) the two triple points forming the ends of the λ line in the phase diagram for helium, where, at the lower end of this line, the gas, normal fluid and superfluid coexist, and, at the upper end, the solid coexists with the normal fluid and superfluid. Similarly, the complextemperature phase boundary B of a spin system may include intersection points at which different curves contained in B meet. We denote the number of curves meeting at such an intersection point as nc. Often these can be grouped into pairs, such that two curves meet with equal tangents at the intersection point; these are then regarded as a single branch of a curve passing through this point. In the terminology of algebraic geometry [8], a singular point of an algebraic curve is a multiple (intersection) point of index m if m branches of the curve pass through this point. This is thus an intersection point with nc = 2m curves