arXiv subjects

Victor Matveev

Publications and source records attributed to Victor Matveev.

10 recordsLinked to original sources

arXiv2007

On Properties of the Ising Model for Complex Energy/Temperature and Magnetic Field

We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable $u=e^{-4K}$, where $K=J/(k_BT)$, 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 $h=H/(k_BT)$, these results provide new insights into features of our earlier study of this case. We also consider complex $h=H/(k_BT)$ 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., $μ=e^{iθ}$, 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 ${\cal B}_u$ contains a real line segment extending through part of the physical ferromagnetic interval $0 \le u \le 1$, with a right-hand endpoint $u_{rhe}$ at the temperature for which the Yang-Lee edge singularity occurs at $μ=e^{\pm iθ}$. Conformal field theory arguments are used to relate the singularities at $u_{rhe}$ and the Yang-Lee edge.

cond-mat.stat-mech↗

arXiv1996

Complex-Temperature Phase Diagram of the 1D $Z_6$ Clock Model and its Connection with Higher-Dimensional Models

We determine the exact complex-temperature (CT) phase diagram of the 1D $Z_6$ 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 $Z_6$ clock models.

cond-mat↗

arXiv1996

Some New Results on Complex-Temperature Singularities in Potts Models on the Square Lattice

We report some new results on the complex-temperature (CT) singularities of $q$-state Potts models on the square lattice. We concentrate on the problematic region $Re(a) < 0$ (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 $3 \le q \le 8$, 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) FM phase. Exponents for these singularities are determined; e.g., for $q=3$, we find $\beta_e=-0.125(1)$, consistent with $\beta_e=-1/8$. By duality, these results also imply associated arcs extending to the (CT extension of the) symmetric PM 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 $a_e,a_e^*=2(-1 \mp i)$. Further discussions of complex-temperature phase diagrams are given.

cond-mat↗

arXiv1995

Some New Results on Yang-Lee Zeros of the Ising Model Partition Function

We prove that for the Ising model on a lattice of dimensionality $d \ge 2$, the zeros of the partition function $Z$ in the complex $μ$ plane (where $μ=e^{-2βH}$) lie on the unit circle $|μ|=1$ for a wider range of $K_{n n'}=βJ_{nn'}$ than the range $K_{n n'} \ge 0$ assumed in the premise of the Yang-Lee circle theorem. This range includes complex temperatures, and we show that it is lattice-dependent. Our results thus complement the Yang-Lee theorem, which applies for any $d$ and any lattice if $J_{nn'} \ge 0$. For the case of uniform couplings $K_{nn'}=K$, we show that these zeros lie on the unit circle $|μ|=1$ not just for the Yang-Lee range $0 \le u \le 1$, but also for (i) $-u_{c,sq} \le u \le 0$ on the square lattice, and (ii) $-u_{c,t} \le u \le 0$ on the triangular lattice, where $u=z^2=e^{-4K}$, $u_{c,sq}=3-2^{3/2}$, and $u_{c,t}=1/3$. For the honeycomb, $3 \cdot 12^2$, and $4 \cdot 8^2$ lattices we prove an exact symmetry of the reduced partition functions, $Z_r(z,-μ)=Z_r(-z,μ)$. This proves that the zeros of $Z$ for these lattices lie on $|μ|=1$ for $-1 \le z \le 0$ as well as the Yang-Lee range $0 \le z \le 1$. Finally, we report some new results on the patterns of zeros for values of $u$ or $z$ outside these ranges.

cond-mat↗

arXiv1995

Complex-Temperature Properties of the 2D Ising Model for Nonzero Magnetic Field

We study the complex-temperature phase diagram of the square-lattice Ising model for nonzero external magnetic field $H$, i.e. for $0 \le \mu \le \infty$, where $\mu=e^{-2\beta H}$. We also carry out a similar analysis for $-\infty \le \mu \le 0$. The results for the interval $-1 \le \mu \le 1$ provide a new way of continuously connecting the two known exact solutions of this model, viz., at $\mu=1$ (Onsager, Yang) and $\mu=-1$ (Lee and Yang). Our methods include calculations of complex-temperature zeros of the partition function and analysis of low-temperature series expansions. For real nonzero $H$, the inner branch of a lima\c{c}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 $\mu < 0$, 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 $\mu=-1$ on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero-field phase diagrams.

cond-mat↗

arXiv1995

A Connection Between Complex-Temperature Properties of the 1D and 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 $θ_n = (1+2n)π/(4s^2)$, 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.

hep-lat↗

arXiv1995

Zeros of the Partition Function for Higher--Spin 2D Ising Models

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 $4[s^2]-2$ such arcs for $s \ge 1$, where $[x]$ denotes the integral part of $x$.

hep-lat↗

arXiv1995

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.

hep-lat↗

arXiv1994

Complex-Temperature Properties of the 2D Ising Model with $βH = \pm i π/2$

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 $βH = \pm i π/2$. 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 $χ$ to $O(u^{23})$, where $u=e^{-4K}$. Analysing this series, we conclude that $χ$ has divergent singularities (i) at $u=u_e=-(3-2^{3/2})$ with exponent $γ_e'=5/4$, (ii) at $u=1$, with exponent $γ_1'=5/2$, and (iii) at $u=u_s=-1$, with exponent $γ_s'=1$. 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.

hep-lat↗

arXiv1994

Complex-Temperature Singularities in the $d=2$ Ising Model. II. Triangular Lattice

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 $\barχ$ 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 $χ$ at $u=u_e=-1/3$ (where $u = e^{-4K}$) is $γ_e'=5/4$ and refine his estimate of the critical amplitude. We discuss the remarkable nature of this singularity, at which the spontaneous magnetisation diverges (with exponent $β_e=-1/8$) 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 $α_e'=1$, so that the relation $α_e'+2β_e+γ_e'=2$ is satisfied. We also study the singularity at $u=u_s=-1$, where $M$ vanishes (with $β_s=3/8$) and $C$ diverges logarithmically (with $α_s' = α_s = 0$).

hep-lat↗