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.
We present calculations of the complex-temperature zeros of the partition functions for 2D Ising models on the square lattice with spin
s=1, 3/2, and 2. These give insight into complex-temperature phase diagrams of these models in the thermodynamic limit. Support is adduced for a conjecture that all divergences of the magnetisation occur at endpoints of arcs of zeros protruding into the FM phase. We conjecture that there are
4[s2]−2 such arcs for
s≥1, where
[x] denotes the integral part of
x.
We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable
u=e−4K, where
K=J/(kBT), for nonzero external magnetic field,
H. Exact results are given for the phase diagram in the
u plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real
h=H/(kBT), these results provide new insights into features of our earlier study of this case. We also consider complex
h=H/(kBT) and
μ=e−2h. Calculations of complex-
u zeros of the partition function on sections of the square lattice are presented. For the case of imaginary
h, i.e.,
μ=eiθ, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the
u plane. We find that in this case, the phase boundary
Bu contains a real line segment extending through part of the physical ferromagnetic interval
0≤u≤1, with a right-hand endpoint
urhe at the temperature for which the Yang-Lee edge singularity occurs at
μ=e±iθ. Conformal field theory arguments are used to relate the singularities at
urhe and the Yang-Lee edge.
We study the complex-temperature phase diagram of the square-lattice Ising model for nonzero external magnetic field H, i.e., for 0\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}\ensuremath{\infty}, where \ensuremath{\mu}=${\mathit{e}}^{\mathrm{\ensuremath{-}}2\mathrm{\ensuremath{\beta}}\mathit{H}}$. We also carry out a similar analysis for -\ensuremath{\infty}\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}0. The results for the interval -1\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1 provide a way of continuously connecting the two known exact solutions of this model, viz., at \ensuremath{\mu}=1 (Onsager, Yang) and \ensuremath{\mu}=-1 (Lee and Yang). Our methods include numerical calculations of complex-temperature zeros of the partition function and an analysis of low-temperature series expansions. For real nonzero H, the inner branch of a lima\ifmmode \mbox{\c{c}}\else \c{c}\fi{}on bounding the FM phase breaks and forms two complex-conjugate arcs. We study the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs. For \ensuremath{\mu}0, there are two line segments of singularities on the negative and positive u axis, and we carry out a similar study of the behavior at the inner endpoints of these arcs, which constitute the nearest singularities to the origin in this case. Finally, we also determine the exact complex-temperature phase diagrams at \ensuremath{\mu}=-1 on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero-field phase diagrams. \textcopyright{} 1996 The American Physical Society.
We study dimensional crossover in Ising systems at complex temperatures by comparing three types of system: the infinite isotropic 2D Ising model; the infinite anisotropic 2D Ising model; and Ising ladders with a finite number of legs. In particular we present evidence, from both tensor-network calculations and numerical evaluations based on the exact solution of the model, that the infinite anisotropic 2D Ising model exhibits long-range spatially modulated magnetization in certain regions of the complex-temperature plane. We discuss the physics of the special unitary points that exists in the complex-temperature plane, and their connections to the theory of quantum information processing.
We study Yang-Lee zeros as they approach the thermodynamic limit of the two-dimensional (2D) nearest-neighbor antiferromagnetic Ising model on square and triangular lattices. We employ the numerical linked cluster expansion (NLCE) equipped with exact enumeration (EE) of the partition function to estimate the Laplacian of the free energy, which is proportional to the zeros density. Using a modified NLCE, where the expansion can be carried directly on the Yang-Lee zeros of the involved clusters, we estimate the density of Yang-Lee zeros in the thermodynamic limit. NLCE gives significantly more zeros than EE in the complex field plane providing more insights on how the root curves look in the thermodynamic limit. For the square lattice at $T\ensuremath{\ll}{T}_{c}$, the results suggest that two vertical lines at $\ifmmode\pm\else\textpm\fi{}{h}_{c}(T)$ in the complex field plane (i.e., two concentric circles in the complex fugacity plane) are the thermodynamic root curves. A similar picture is found for the triangular lattice for phase transitions at large values of magnetic field while further study is needed for phase transitions at smaller values of magnetic field. The convergence of the NLCE and (EE) calculations of the partition function to the thermodynamic limit is studied in both lattices and the temperature-field phase diagram is obtained from Yang-Lee zeros using both methods. This NLCE-based approach will facilitate the study of different types of phase transitions using Yang-Lee zeros in future research.
We study the complex-temperature properties of a rare example of a statistical mechanical model which is exactly solvable in an external symmetry-breaking field, namely, the Ising model on the square lattice with beta H=+or-i pi /2. This model was solved by Lee and Yang (1952). We first determine the complex-temperature phases and their boundaries. From a low-temperature, high-field series expansion of the partition function, we extract the low-temperature series for the susceptibility chi to O(u 23 ), where u=e -4K . Analysing this series, we conclude that chi has divergent singularities (i) at u=u 3 =-(3-2 3 2/) with exponent gamma e '=5/4, (ii) at u=1, with exponent gamma 1 '=5/2, and (iii) at u=u s =-1, with exponent gamma s '=1. We also extract a shorter series for the staggered susceptibility and investigate its singularities. Using the exact result of Lee and Yang for the free energy, we calculate the specific heat and determine its complex-temperature singularities. We also carry this out for the uniform and staggered magnetization.