We investigate the spin-spin correlation functions of Ising magnets at complex values of the temperature, T. For one-dimensional chain and ladder systems, we show the existence of a kind of helimagnetic order in the vicinity of contours where the leading two eigenvalues of the transfer matrix become equal in magnitude. We analyse the development of long-range order as the two-dimensional limit is approached, and find that there is rich structure in much of the complex-T plane. In particular, and contrary to the work of Fisher on this problem, the development of long-range order is actually associated with a proliferation of partition function zeros in a certain finite region of that plane containing the real-temperature magnetically ordered phase. The thermodynamic consequences of this are also discussed.
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.
As is well known, one-dimensional spin systems with short-range interactions cannot undergo a phase transition at finite temperature. However, in many 1D Ising spin models with frustration, at a certain crossover point, the thermodynamic quantities show anomalies that resemble such a transition W. Yin [Phys. Rev. Res. 6, 013331 (2024)2643-156410.1103/PhysRevResearch.6.013331]. Recently, Yin [Phys. Rev. Res. 6, 013331 (2024)2643-156410.1103/PhysRevResearch.6.013331] has shown, using decorated ladders as an example, that a one-dimensional Ising model can be arbitrarily close to the genuine phase transition. Using the transfer matrix method and real space renormalization group technique, we find that crossover resembling a phase transition accompanied by gigantic specific heat can also occur in two-dimensional systems. We consider coupled Ising spin chains with a zig-zag structure and a four-spin j_{4} interaction, which allows us to control the temperature at which the spins become unbound in the model with short-range interactions. The crossover point is defined by two parameters: the crossover temperature t=t^{*} and the crossover interaction j_{4}=j_{4}^{*}, and occurs when the temperature at which the effective interaction between the spin subsystems zeros out coincides with the temperature at which the spins become unbound. In systems in which a genuine phase transition occurs at the critical temperature t=t_{c}, there are two possibilities: if t_{c}>t^{*} the system undergoes a sequence of three phase transitions with a reentrant transition to disordered phase, while for t_{c}<t^{*} the phase transition is preceded by the considered crossover. We study the key role in the appearance of anomalies in the thermodynamic quantities of the four-spin interactions which are suspected to have a significant impact on the properties of magnetic two-dimensional lattices, but possible also on the construction of social systems modeling the phenomenon of social validation or peer pressure.
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.
The Ising model describes collective behaviors such as phase transitions and critical phenomena in various physical, biological, economical, and social systems. It is well known that spontaneous phase transition at finite temperature does not exist in the Ising model with short-range interactions in one dimension. Yet, little is known about whether this forbidden phase transition can be approached arbitrarily closely—at fixed finite temperature. Here I use symmetry analysis of the transfer matrix to reveal the existence of spontaneous ultranarrow phase crossover (UNPC) at finite temperature in one class of one-dimensional Ising models on decorated two-leg ladders, in which the crossover temperature T0 is determined solely by on-rung interactions and decorations, while the crossover width 2δT is independently, exponentially reduced ( δT=0 means a genuine phase transition) by on-leg interactions and decorations. These findings establish a simple ideal paradigm for realizing an infinite number of one-dimensional Ising systems with spontaneous UNPC at desirable T0 , which would be characterized in routine laboratory measurements as a genuine first-order phase transition with large latent heat thanks to the ultranarrow δT (say less than one nanokelvin), paving a way to push the limit in our understanding of phase transitions and the dynamical actions of frustration arbitrarily close to the forbidden regime. Published by the American Physical Society 2024
Monte Carlo calculations have been used to study the behavior of a simple cubic spin-\textonehalf{} Ising model with spatially anisotropic nearest-neighbor coupling. The interaction in the
x and
y directions, which is equal to (${J}_{\ensuremath{\perp}}$), was systematically decreased compared to that in the
z direction (${J}_{\ensuremath{\parallel}}$), and results were obtained for values of $\ensuremath{\Delta}=\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}$ as small as 0.003. For small $\ensuremath{\Delta}$ the system behaves like a one-dimensional Ising chain at high temperature before undergoing a low-temperature transition to a three-dimensionally ordered state. We find that the asymptotic ($\ensuremath{\Delta}\ensuremath{\rightarrow}0$) variation of
Tc is accurate over a surprisingly wide range of $\ensuremath{\Delta}$.