Partition function zeros for aperiodic systems

The study of zeros of partition functions, initiated by Yang and Lee, provides an important qualitative and quantitative tool in the study of critical phenomena. This has frequently been used for periodic as well as hierarchical lattices. Here, we consider magnetic field and temperature zeros of Ising model partition functions on several aperiodic structures. In 1D, we analyze aperiodic chains obtained from substitution rules, the most prominent example being the Fibonacci chain. In 2D, we focus on the tenfold symmetric triangular tiling which allows efficient numerical treatment by means of corner transfer matrices.     

On the Distribution and Gap Structure of Lee–Yang Zeros for the Ising Model: Periodic and Aperiodic Couplings

In this work, we present some results on the distribution of Lee–Yang zeros for the ferromagnetic Ising model on the rooted Cayley Tree (Bethe Lattice), assuming free boundary conditions, and in the one-dimensional lattice with periodic boundary conditions. In the case of the Cayley Tree, we derive the conditions that the interactions between spins must obey in order to ensure existence or absence of phase transition at finite temperature (T0). The results are first obtained for periodic interactions along the generations of the lattice. Then, using periodic approximants, we are also able to obtain results for aperiodic sequences generated by substitution rules acting on a finite alphabet. The particular examples of the Fibonacci and the Thue-Morse sequences are discussed. Most of the results are obtained for a Cayley Tree with arbitrary order d. We will be concerned in showing whether or not the zeros become dense in the whole unit circle of the fugacity variable. Regarding the one-dimensional Ising model, we derive a general treatment for the structure of gaps (regions free of Lee–Yang zeros) around the unit circle.     

Statistical mechanics of a quasiperiodic one-dimensional magnetic chain

In this paper we study the thermodynamic properties of the Ising model on a linear chain in which the sites are generated according to the Fibonacci sequence. We calculate the partition function, the specific heat and the q-dependent magnetic susceptibility.     

Fibonacci序列的统计属性和电子输运系数

对按膨胀规律A→AB和B→A生成的Fibonacci序列,采用一维随机行走模型数值计算了序列的自相关函数以及自行定义的准标准偏差.利用Hurst分析法研究了序列的再标度范围函数及其Hurst指数,并将结果与一维随机二元序列进行了对比.发现这些统计量有奇特的准周期振荡行为以及小于05的Hurst指数,直接论证了Fibonacci序列具有关联、标度不变及自相似等性质.从Anderson紧束缚模型出发,采用传输矩阵方法研究了Fibonacci序列的电子输运特性,讨论了输运系数对能量及其序列长度的依赖关系.研究 关键词： Fibonacci序列 统计属性 电子输运系数     

General theory of lee-yang zeros in models with first-order phase transitions

We present a general, rigorous theory of Lee-Yang zeros for models with first-order phase transitions that admit convergent contour expansions. We derive formulas for the positions and the density of the zeros. In particular, we show that, for models without symmetry, the curves on which the zeros lie are generically not circles, and can have topologically nontrivial features, such as bifurcation. Our results are illustrated in three models in a complex field: the low-temperature Ising and Blume-Capel models, and the q-state Potts model for large q.     

Lee–Yang Theorems and the Complexity of Computing Averages

We study the complexity of computing average quantities related to spin systems, such as the mean magnetization and susceptibility in the ferromagnetic Ising model, and the average dimer count (or average size of a matching) in the monomer-dimer model. By establishing connections between the complexity of computing these averages and the location of the complex zeros of the partition function, we show that these averages are #P-hard to compute, and hence, under standard assumptions, computationally intractable. In the case of the Ising model, our approach requires us to prove an extension of the famous Lee–Yang Theorem from the 1950s.     

Random matrix theory for pseudo-Hermitian systems: Cyclic blocks

We discuss the relevance of random matrix theory for pseudo-Hermitian systems, and, for Hamiltonians that break parity P and time-reversal invariance T. In an attempt to understand the random Ising model, we present the treatment of cyclic asymmetric matrices with blocks and show that the nearest-neighbour spacing distributions have the same form as obtained for the matrices with scalar entries. We also summarize the theory for random cyclic matrices with scalar entries. We have also found that for block matrices made of Hermitian and pseudo-Hermitian sub-blocks of the form appearing in Ising model depart from the known results for scalar entries. However, there is still similarity in trends even in log-log plots.     

Yang-Lee zeros of the antiferromagnetic Ising model

There exists the famous circle theorem on the Yang-Lee zeros of the ferromagnetic Ising model. However, the Yang-Lee zeros of the antiferromagnetic Ising model are much less well understood than those of the ferromagnetic model. The precise distribution of the Yang-Lee zeros of the antiferromagnetic Ising model only with nearest-neighbor interaction J on LxL square lattices is determined as a function of temperature a=e(2betaJ) (J<0), and its relation to the phase transitions is investigated. In the thermodynamic limit (L-->infinity), the distribution of the Yang-Lee zeros of the antiferromagnetic Ising model cuts the positive real axis in the complex x=e(-2betaH) plane, resulting in the critical magnetic field +/-H(c)(a), where H(c)>0 below the critical temperature a(c)=square root of 2-1. The results suggest that the value of the scaling exponent y(h) is 1 along the critical line for a相似文献   

Optical transmission spectra in symmetrical Fibonacci photonic multilayers

We study the transmission properties of light through the symmetric Fibonacci photonic multilayers, i.e, a binary one-dimensional quasiperiodic structure, made up of both positive (SiO₂) and negative refractive index materials with a mirror symmetry. These spectra are calculated by using a theoretical model based on the transfer matrix approach for normal incidence geometry, in which many perfect transmission peaks (the transmission coefficients are equal to the unity) are numerically obtained. Besides, the transmission coefficient exhibits a six-cycle self-similar behavior with respect to the generation number of the Fibonacci sequence.     

光学波段菲波纳契序列一维光子晶体纳米膜传输特性研究

为了研究光学波段菲波纳契序列一维光子晶体纳米膜的传输特性,应用传输矩阵方法数值模拟各种情况下的透射率即传输函数随频率的变化.数值结果表明在正入射时,菲波纳契序列一维光子晶体中的禁带宽度、中心位置、数目都与构成序列的项数、组元物理厚度、组成序列组元初始次序、组元折射率差值都对传输特性有较大影响,在可见光区组元折射率差值越大越易形成较宽禁带,进一步研究广义菲波纳契序列一维光子晶体纳米膜的传输特性,发现比典型情况更易在可见光区形成禁带.     

Omnidirectional Single-negative Gap and in Fibonacci Sequences Composed of Single-negative Materials

The band structures of Fibonacci sequence composed of single-negative materials are studied with a transfer matrix method. A new type of omnidirectional single-negative gaps is found in the Fibonacci sequence. In contrast to the Bragg gaps, such an omnidirectional single-negative gap is insensitive to the incident angles and polarization, and is invariant upon the change of the ratio of the thicknesses of two media. It is found that omnidirectional single-negative gap exists in the other Fibonacci sequence, and it is rather stable and independence of the structure sequence.     

H = xp model revisited and the Riemann zeros

Berry and Keating conjectured that the classical Hamiltonian H = xp is related to the Riemann zeros. A regularization of this model yields semiclassical energies that behave, on average, as the nontrivial zeros of the Riemann zeta function. However, the classical trajectories are not closed, rendering the model incomplete. In this Letter, we show that the Hamiltonian H = x(p + ?(p)2/p) contains closed periodic orbits, and that its spectrum coincides with the average Riemann zeros. This result is generalized to Dirichlet L functions using different self-adjoint extensions of H. We discuss the relation of our work to Polya's fake zeta function and suggest an experimental realization in terms of the Landau model.     

Theory of the Ising model on a Cayley tree

The Ising model on a Cayley tree is known to exhibit a phase transition of continuous order. In this paper we present a complete and quantitative analysis of the leading singular term in the free energy which is associated with this phase transition. We have been able to solve this problem by considering the distribution of zeros of the partition function. The most interesting new feature in our results is a contribution to the free energy which performs singular oscillations as the magnetic field approaches zero.     

Yang–Lee zeros of the Ising model on random graphs of non planar topology

We obtain in a closed form the 1/N² contribution to the free energy of the two Hermitian N×N random matrix model with nonsymmetric quartic potential. From this result, we calculate numerically the Yang–Lee zeros of the 2D Ising model on dynamical random graphs with the topology of a torus up to n=16 vertices. They are found to be located on the unit circle on the complex fugacity plane. In order to include contributions of even higher topologies we calculated analytically the nonperturbative (sum over all genus) partition function of the model for the special cases of N=1,2 and graphs with n≤20 vertices. Once again the Yang–Lee zeros are shown numerically to lie on the unit circle on the complex fugacity plane. Our results thus generalize previous numerical results on random graphs by going beyond the planar approximation and strongly indicate that there might be a generalization of the Lee–Yang circle theorem for dynamical random graphs.     

Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators

Simon’s subshift conjecture states that for every aperiodic minimal subshift of Verblunsky coefficients, the common essential support of the associated measures has zero Lebesgue measure. We disprove this conjecture in this paper, both in the form stated and in the analogous formulation of it for discrete Schrödinger operators. In addition we prove a weak version of the conjecture in the Schrödinger setting. Namely, under some additional assumptions on the subshift, we show that the density of states measure, a natural measure associated with the operator family and whose topological support is equal to the spectrum, is singular. We also consider one-frequency quasi-periodic Schrödinger operators with continuous sampling functions and show that generically, the density of states measure is singular as well.     

Auxiliary Vertices Method for Kagomé-Lattice Eight-Vertex Model

We investigate a class of eight-vertex models on a Kagomé lattice. With the help of auxiliary vertices, the Kagomé-lattice eight-vertex model (KEVM) is related to an inhomogeneous system which leads to a one-parameter family of commuting transfer matrices. Using an equation for commuting transfer matrices, we determine their eigenvalues. From calculated eigenvalues the correlation length of the KEVM is derived with its full anisotropy. There are two cases: In the first case the anisotropic correlation length (ACL) is the same as that of the triangular/honeycomb-lattice Ising model. By the use of an algebraic curve, it is shown that the Kagomé-lattice Ising model, the diced-lattice Ising model, and the hard-hexagon model also have (essentially) the same ACL as the KEVM. In the second case we find that the ACL displays 12fold rotational symmetry.     

Non-Abelian braiding of lattice bosons

We report on a numerical experiment in which we use time-dependent potentials to braid non-Abelian quasiparticles. We consider lattice bosons in a uniform magnetic field within the fractional quantum Hall regime, where ν, the ratio of particles to flux quanta, is near 1/2, 1, or 3/2. We introduce time-dependent potentials which move quasiparticle excitations around one another, explicitly simulating a braiding operation which could implement part of a gate in a quantum computation. We find that different braids do not commute for ν near 1 and 3/2, with Berry matrices, respectively, consistent with Ising and Fibonacci anyons. Near ν=1/2, the braids commute.     

The oscillatory behavior of the high-temperature expansion of Dyson's hierarchical model: A renormalization group analysis

We calculate 800 coefficients of the high-temperature expansion of the magnetic susceptibility of Dyson's hierarchical model with a Landau-Ginzburg measure. Log-periodic corrections to the scaling laws appear as in the case of an Ising measure. The period of oscillation appears to be a universal quantity given in good approximation by the logarithm of the largest eigenvalue of the linearized RG transformation, in agreement with a possibility suggested by Wilson and developed by Niemeijer and van Leeuwen. We estimate to be 1.300 (with a systematic error of the order of 0.002), in good agreement with the results obtained with other methods, such as the -expansion. We briefly discuss the relationship between the oscillations and the zeros of the partition function near the critical point in the complex temperature plane.     

Yang-Lee zeros of triangular Ising antiferromagnets

In our previous research, by combining both the exact enumeration method (microcanonical transfer matrix) for a small system (L=9) with the Wang-Landau Monte Carlo algorithm for large systems (to L=30) we obtained the exact and approximate densities of states g(M,E), as a function of the magnetization M and exchange energy E, for a triangular-lattice Ising model. In this paper, based on the density of states g(M,E), the precise distribution of the Yang-Lee zeros of triangular-lattice Ising antiferromagnets is obtained in a uniform magnetic field as a function of temperature for a 9×9 lattice system. Also, the feasibility of the Yang-Lee zero approach combined with the Wang-Landau algorithm is demonstrated; as a result, we obtained the magnetic exponents for triangular Ising antiferromagnets at various temperatures.