Complex-temperature Singularities Research Articles

Partition function zeros of the Q-state Potts model on the simple-cubic lattice

The Q-state Potts model on the simple-cubic lattice is studied using the zeros of the exact partition function on a finite lattice. The critical behavior of the model in the ferromagnetic and antiferromagnetic phases is discussed based on the distribution of the zeros in the complex temperature plane. The characteristic exponents at complex-temperature singularities, which coexist with the physical critical points in the complex temperature plane for no magnetic field ( H q =0), are estimated using the low-temperature series expansion. We also study the partition function zeros of the Potts model for nonzero magnetic field. For H q >0 the physical critical points disappear and the Fisher edge singularities appear in the complex temperature plane. The characteristic exponents at the Fisher edge singularities are calculated using the high-field, low-temperature series expansion. It seems that the Fisher edge singularity is related to the Yang–Lee edge singularity which appears in the complex magnetic-field plane for T> T c .

Extremely inhomogeneous states in multifractal thermodynamics of particle collisions at high energies

Extremely inhomogeneous states in the rapidity space of multiparticle production processes are studied using an analytic continuation based on complex-temperature singularities in thermodynamics of multifractals. Good agreement between this approach and results of laboratory experiments on hadron and nuclear collisions at high energies is established.

Complex-temperature singularities and their application to the thermodynamic description of multifractals

It is shown that applicability of standard high-temperature expansions in multifractal thermodynamics is restricted by complex-temperature singularities. An analytic continuation method (finite-temperature expansions) has been developed to improve the analytic expansions approach. The quantum intermittency is then used as an example of applicability of the finite-temperature expansion.

Singularities at a dense set of temperature singularities in the Husimi tree

We investigate complex temperature singularities of the three-site interacting Ising model on the Husimi tree in the presentce of magnetic field. We show that at certain magnetic field these singularities lie at a dense set and as a consequence the phase transition condensation take place.

Multifractality at Transition to Chaos Complex-Temperature Singularities

It is shown, that multifractal complex-temperature singularities can play a significant role in the critical strange sets multifractality. These singularities lead to a finite radius of convergence of the real high-temperature expansions and, therefore, to necessity to use a finite-temperature expansions (an analytic continuation). It is shown, using analytic results on multifractality of strange attractors of the baker map and results of numerical computations of the multifractal spectra on all critical points of phase transitions from period-η-tupling to chaos in 1D iterative system (Chinese Phys. Lett.3, 285 (1986) and J. Phys.A25, 589 (1992)) as well as results of a recent numerical simulation of a quantum system with multifractal spectrum (J. Phys.A28, 2717 (1995)), that the finite-temperature expansions give good approximation for the generalized dimensions Dq in a representative interval of q.

Complex-temperature singularities in Potts models on the square lattice.

We report some results on the complex-temperature (CT) singularities of $q$-state Potts models on the square lattice. We concentrate on the problematic region $\mathrm{Re}(a)<0$ (where $a={e}^{K}$) in which CT zeros of the partition function are sensitive to finite lattice artifacts. From analyses of low-temperature series expansions for $3<~q<~8$, we establish the existence, in this region, of complex-conjugate CT singularities at which the magnetization and susceptibility diverge. From calculations of zeros of the partition function, we obtain evidence consistent with the inference that these singularities occur at endpoints ${a}_{e}$,${a}_{e}^{*}$ of arcs protruding into the (complex-temperature extension of the) ferromagnetic phase. Exponents for these singularities are determined; e.g., for $q=3$, we find ${\ensuremath{\beta}}_{e}=\ensuremath{-}0.125(1)$, consistent with ${\ensuremath{\beta}}_{e}=\ensuremath{-}\frac{1}{8}$. By duality, these results also imply associated arcs extending into the (CT extension of the) symmetric paramagnetic phase. Analytic expressions are suggested for the positions of some of these singularities; e.g., for $q=5$, our finding is consistent with the exact value ${a}_{e}$, ${a}_{e}^{*}=2(\ensuremath{-}1\ensuremath{\mp}i)$. Further discussions of complex-temperature phase diagrams are given.

Complex-temperature singularities of Ising models

We report new results on complex-temperature properties of Ising models. These include studies of the $s=1/2$ model on triangular, honeycomb, kagom\'e, $3 \cdot 12^2$, and $4 \cdot 8^2$ lattices. We elucidate the complex--$T$ phase diagrams of the higher-spin 2D Ising models, using calculations of partition function zeros. Finally, we investigate the 2D Ising model in an external magnetic field, mapping the complex--$T$ phase diagram and exploring various singularities therein. For the case $\beta H=i\pi/2$, we give exact results on the phase diagram and obtain susceptibility exponents $\gamma'$ at various singularities from low-temperature series analyses.

Complex-temperature singularities in thed= 2 Ising model: triangular and honeycomb lattices

We investigate complex-temperature singularities in the Ising model on the triangular lattice. Extending an earlier analysis of the low-temperature series expansions for the (zero-field) susceptibility $\bar\chi$ by Guttmann \cite{g75} to include the use of differential approximants, we obtain further evidence in support of his conclusion that the exponent describing the divergence in $\chi$ at $u=u_e=-1/3$ (where $u = e^{-4K}$) is $\gamma_e'=5/4$ and refine his estimate of the critical amplitude. We discuss the remarkable nature of this singularity, at which the spontaneous magnetisation diverges (with exponent $\beta_e=-1/8$) and show that it lies at the endpoint of a singular line segment constituting part of the natural boundaries of the free energy in the complex $u$ plane. Using exact results, we find that the specific heat has a divergent singularity at $u=-1/3$ with exponent $\alpha_e'=1$, so that the relation $\alpha_e'+2\beta_e+\gamma_e'=2$ is satisfied. We also study the singularity at $u=u_s=-1$, where $M$ vanishes (with $\beta_s=3/8$) and $C$ diverges logarithmically (with $\alpha_s' = \alpha_s = 0$).

Complex-temperature properties of the Ising model on 2D heteropolygonal lattices

Using exact results, we determine the complex-temperature phase diagrams of the 2D Ising model on three regular heteropolygonal lattices, $(3 \cdot 6 \cdot 3 \cdot 6)$ (kagom\'{e}), $(3 \cdot 12^2)$, and $(4 \cdot 8^2)$ (bathroom tile), where the notation denotes the regular $n$-sided polygons adjacent to each vertex. We also work out the exact complex-temperature singularities of the spontaneous magnetisation. A comparison with the properties on the square, triangular, and hexagonal lattices is given. In particular, we find the first case where, even for isotropic spin-spin exchange couplings, the nontrivial non-analyticities of the free energy of the Ising model lie in a two-dimensional, rather than one-dimensional, algebraic variety in the $z=e^{-2K}$ plane.

Complex-temperature properties of the 2D Ising model with beta H=+or-i pi /2

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 $\beta H = \pm i \pi/2$. This model was solved by Lee and Yang \cite{ly}. 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 $u=u_e=-(3-2^{3/2})$ 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 magnetisation.

Complex-temperature singularities of the susceptibility in the d = 2 Ising model

We report new results on complex-temperature singularities of the susceptibility χ of the 2D Ising model. We give a unified treatment of both the physical critical point and the singularity at u = u 3 = −1, which yields the relation γ′ s = 2(γ′ − 1). Using exact results, we show that universality is violated at the singularity u = −1 and discuss the reasons. We also show that certain exponent relations are violated at this point. The behavior of χ is analyzed as u = −1 is approached from the symmetric PM phase and from the FM and AFM phases.

Complex-temperature singularities of the susceptibility in the D=2 Ising model. I. Square lattice

We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that as one approaches the point $u=u_s=-1$ (where $u=e^{-4K}$) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form $\chi \sim A_s'(1+u)^{-\gamma_s'}$ with exponent $\gamma_s'=3/2$. The critical amplitude $A_s'$ is calculated. Other critical exponents are found to be $\alpha_s'=\alpha_s=0$ and $\beta_s=1/4$, so that the scaling relation $\alpha_s'+2\beta_s+\gamma_s'=2$ is satisfied. However, using exact results for $\beta_s$ on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: $\beta_s$ is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point $u=-1$ from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.

Complex temperature singularities for the two-dimensional Heisenberg O(∞) model

We compute the complex temperature singularities for the two-dimensional O( N) symmetric Heisenberg model in the limit N → ∞. The model is exactly soluble but the analysis of singularities requires a quite arduous study of analytic properties of the inverse of an elliptic function with respect to its modulus. We show that the mass gap and other physical quantities are analytic in a narrow strip around the real inverse temperature axis. This fact explains why low-temperature properties are not easily described by high-temperature expansion methods. We then use the known analyticity properties to identify a conformal transformation which improves the convergence of the high-temperature expansion and, as a consequence, the estimate of physical quantities at low temperature. The method is suitable for extension to all N > 2 models, in which a similar analytic structure presumably holds.