Critical Two-dimensional Ising Model Research Articles

Bond-weighting method for the Grassmann tensor renormalization group

Recently, the tensor network description with bond weights on its edges has been proposed as a novel improvement for the tensor renormalization group algorithm. The bond weight is controlled by a single hyperparameter, whose optimal value is estimated in the original work via the numerical computation of the two-dimensional critical Ising model. We develop this bond-weighted tensor renormalization group algorithm to make it applicable to the fermionic system, benchmarking with the two-dimensional massless Wilson fermion. We show that the accuracy with the fixed bond dimension is improved also in the fermionic system and provide numerical evidence that the optimal choice of the hyperparameter is not affected by whether the system is bosonic or fermionic. In addition, by monitoring the singular value spectrum, we find that the scale-invariant structure of the renormalized Grassmann tensor is successfully kept by the bond-weighting technique.

The two-dimensional continuum random field Ising model

In this paper, we construct the two-dimensional continuum random field Ising model via scaling limits of a random field perturbation of the critical two-dimensional Ising model with diminishing disorder strength. Furthermore, we show that almost surely with respect to the continuum random field given by a white noise, the law of the magnetisation field is singular with respect to that of the two-dimensional continuum pure Ising model constructed by Camia, Garban and Newman (Ann. Probab. 43 (2015) 528–571).

A CLT for the total energy of the two-dimensional critical Ising model

Consider the Ising model on $([1,2N]\times[1,2M])\cap\mathbb{Z}^2$ at critical temperature with periodic boundary condition in the horizontal direction and free boundary condition in the vertical direction. Let $E_{M,N}$ be its total energy (or Hamiltonian). Suppose $M$ is a function of $N$ satisfying $M\geq N/(\ln N)^{\alpha}$ for some $\alpha\in[0,1)$. In particular, one may take $M=N$. We prove that \begin{equation*} \frac{E_{M,N}+4\sqrt{2}M N-(4/\pi)N\ln N}{\sqrt{(32/\pi)MN\ln N}} \end{equation*} converges weakly to a standard Gaussian distribution as $N\rightarrow\infty$.

Exact Logarithmic Four-Point Functions in the Critical Two-Dimensional Ising Model.

Based on conformal symmetry we propose an exact formula for the four-point connectivities of Fortuin-Kasteleyn clusters in the critical Ising model when the four points are anchored to the boundary. The explicit solution we found displays logarithmic singularities. We check our prediction using MonteCarlo simulations on a triangular lattice, showing excellent agreement. Our findings could shed further light on the formidable task of the characterization of logarithmic conformal field theories and on their relevance in physics.

Numerical estimation of structure constants in the three-dimensional Ising conformal field theory through Markov chain uv sampler.

Herdeiro and Doyon [Phys. Rev. E 94, 043322 (2016)2470-004510.1103/PhysRevE.94.043322] introduced a numerical recipe, dubbed uv sampler, offering precise estimations of the conformal field theory (CFT) data of the planar two-dimensional (2D) critical Ising model. It made use of scale invariance emerging at the critical point in order to sample finite sublattice marginals of the infinite plane Gibbs measure of the model by producing holographic boundary distributions. The main ingredient of the Markov chain Monte Carlo sampler is the invariance under dilation. This paper presents a generalization to higher dimensions with the critical 3D Ising model. This leads to numerical estimations of a subset of the CFT data-scaling weights and structure constants-through fitting of measured correlation functions. The results are shown to agree with the recent most precise estimations from numerical bootstrap methods [Kos, Poland, Simmons-Duffin, and Vichi, J. High Energy Phys. 08 (2016) 03610.1007/JHEP08(2016)036].

Markov chain sampling of the O(n) loop models on the infinite plane.

A numerical method was recently proposed in Herdeiro and Doyon [Phys. Rev. E 94, 043322 (2016)10.1103/PhysRevE.94.043322] showing a precise sampling of the infinite plane two-dimensional critical Ising model for finite lattice subsections. The present note extends the method to a larger class of models, namely the O(n) loop gas models for n∈(1,2]. We argue that even though the Gibbs measure is nonlocal, it is factorizable on finite subsections when sufficient information on the loops touching the boundaries is stored. Our results attempt to show that provided an efficient Markov chain mixing algorithm and an improved discrete lattice dilation procedure the planar limit of the O(n) models can be numerically studied with efficiency similar to the Ising case. This confirms that scale invariance is the only requirement for the present numerical method to work.

Self-averaging in the random two-dimensional Ising ferromagnet.

We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size L like ∼Llnln(L). In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a δ peak in the thermodynamic limit L→∞. While previously a lack of self-averaging was found for the free energy, we here obtain results for quantities that are directly measurable in simulations, and implications for measurements in the actual lattice system are discussed.

Observation of Schramm–Loewner evolution on the geometrical clusters of the Ising model

Schramm–Loewner Evolution (SLE) is a stochastic process that, by focusing on the geometrical features of the two-dimensional (2D) conformal invariant models, classifies them using one real parameter κ. In this work we apply the SLE formalism to the exterior frontiers of the geometrical clusters (interfaces) of the two-dimensional critical Ising model on the triangular lattice. We first analyze the critical curves going from the real axis to the real axis in the upper half plane geometry and show numerically that SLE(κ, κ − 6) works well to extract the diffusivity parameter κ. We then analyze the conformal loops of the critical Ising model. After determining some geometrical exponents of the critical loops as the interfaces of the model in hand, we address the problem of application of SLE to conformal loops. We numerically show that SLE(κ, κ − 6) is more reliable than previous methods.

Critical two-dimensional Ising model with free, fixed ferromagnetic, fixed antiferromagnetic, and double antiferromagnetic boundaries.

The critical two-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, and fixed double antiferromagnetic. Using bond propagation algorithms with surface fields, we obtain the free energy, internal energy, and specific heat numerically on square lattices with a square shape and various combinations of the four types of boundary conditions. The calculations are carried out on the square lattices with size N×N and 30<N<1000. The numerical data are analyzed with finite-size scaling. The bulk, edge, and corner terms are extracted very accurately. The exact results are conjectured for the corner logarithmic term in the free energy, the edge logarithmic term in the internal energy, and the corner logarithmic term in the specific heat. The corner logarithmic terms in the free energy agree with the conformal field theory very well.

Accurate Expansions of Internal Energy and Specific Heat of Critical Two-Dimensional Ising Model with Free Boundaries

The bond-propagation algorithm for the specific heat of the two dimensional Ising model is developed and that for the internal energy is completed. Using these algorithms, we study the critical internal energy and specific heat of the model on the square lattice and triangular lattice with free boundaries. Comparing with previous works (Phys Rev E 86:041149, 2012; Phys Rev E 87:022124, 2013), we reach much higher accuracy (\(10^{-28}\)) of the internal energy and specific heat, compared to the accuracy \(10^{-11}\) of the internal energy and \(10^{-9}\) of the specific heat reached in the previous works. This leads to much more accurate estimations of the edge and corner terms. The exact values of all edge and corner terms are therefore conjectured. The accurate forms of finite-size scaling for the internal energy and specific heat are determined for the rectangle-shaped square lattice with various aspect ratios and various shaped triangular lattice. For the rectangle-shaped square and triangular lattices and the triangle-shaped triangular lattice, there is no logarithmic correction terms of order higher than \(1/S\), with \(S\) the area of the system. For the triangular lattice in rhombus, trapezoid and hexagonal shapes, there exist logarithmic correction terms of order higher than \(1/S\) for the internal energy, and logarithmic correction terms of all orders for the specific heat.

Critical Ising model and spanning trees partition functions

We prove that the squared partition function of the two-dimensional critical Ising model defined on a finite, isoradial graph $G=(V,E)$, is equal to $2^{|V|}$ times the partition function of spanning trees of the graph $\bar{G}$, where $\bar{G}$ is the graph $G$ extended along the boundary; edges of $G$ are assigned Kenyon's [Ken02] critical weights, and boundary edges of $\bar{G}$ have specific weights. The proof is an explicit construction, providing a new relation on the level of configurations between two classical, critical models of statistical mechanics.

Shape-dependent finite-size effect of the critical two-dimensional Ising model on a triangular lattice

Using a bond-propagation algorithm, we study the finite-size behavior of the critical two-dimensional Ising model on a finite triangular lattice with free boundaries in five shapes: triangular, rhomboid, trapezoid, hexagonal, and rectangular. The critical free energy, internal energy, and specific heat are calculated. The accuracy of the free energy reaches 10(-26). Based on accurate data on several finite systems with linear size up to N=2000, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat accurately. We confirm the conformal field theory prediction that the corner free energy is universal and find logarithmic corrections in higher-order terms in the critical free energy for the rhomboid, trapezoid, and hexagonal systems, which are absent for the triangular and rectangular systems. The logarithmic edge corrections due to edges parallel or perpendicular to the bond directions in the internal energy are found to be identical, while the logarithmic edge corrections due to corresponding edges in the specific heat are different. The corner internal energy and corner specific heat for angles π/3, π/2, and 2π/3 are obtained, as well as higher-order corrections. Comparing with the corner internal energy and corner specific heat we previously found on a rectangle of the square lattice [Phys. Rev. E 86, 041149 (2012)], we conclude that the corner internal energy and corner specific heat for the rectangular shape are not universal.

Behavior of the two-dimensional Ising model at the boundary of a half-infinite cylinder

The critical two-dimensional Ising model is studied at the boundary of a half-infinite cylinder for the three regular lattices (square, triangular, and hexagonal). The probability of having precisely 2n spinflips at the boundary is computed as a function of the positions, ki, i = 1, 2, 3, …, 2n, of the spinflips. The limit when the mesh goes to zero is obtained. For the square lattice, the probability of having 2n spinflips, independent of their position, is computed at the critical temperature. As a byproduct, we recover De Coninck’s result showing that the limiting distribution of the number of spinflips is Gaussian and we give explicit expressions for its mean and variance. The results are obtained as consequences of Onsager’s solution and are rigorous.

Towards a universal set of topologically protected gates for quantum computation with Pfaffian qubits

We review the topological quantum computation scheme of Das Sarma et al. from the perspective of the conformal field theory for the two-dimensional critical Ising model. This scheme originally used the monodromy properties of the non-Abelian excitations in the Pfaffian quantum Hall state to construct elementary qubits and execute logical NOT on them. We extend the scheme of Das Sarma et al. by exploiting the explicit braiding transformations for the Pfaffian wave functions containing 4 and 6 quasiholes to implement, for the first time in this context, the single-qubit Hadamard and phase gates and the two-qubit Controlled-NOT gate over Pfaffian qubits in a topologically protected way. In more detail, we explicitly construct the unitary representations of the braid groups B 4 , B 6 and B 8 and use the elementary braid matrices to implement one-, two- and three-qubit gates. We also propose to construct a topologically protected Toffoli gate, in terms of a braid-group based Controlled–Controlled-Z gate precursor. Finally we discuss some difficulties arising in the embedding of the Clifford gates and address several important questions about topological quantum computation in general.

Fractal structure of spin clusters and domain walls in the two-dimensional Ising model

The fractal structure of spin clusters and their boundaries in the critical two-dimensional Ising model is investigated numerically. The fractal dimensions of these geometrical objects are estimated by means of Monte Carlo simulations on relatively small lattices through standard finite-size scaling. The obtained results are in excellent agreement with theoretical predictions and partly provide significant improvements in precision over existing numerical estimates.

Short-time critical dynamics of the Ising model studied within the N-fold algorithm

The short-time behaviour of the critical two-dimensional Ising model is studied for the Bortz–Kalos–Lebowitz N-fold way algorithm (BKL algorithm). For square lattices of linear sizes L=110, 128, 140, 170 , and 200, we calculate the short-time critical behaviour for the BKL algorithm and also for the heat-bath algorithm. Comparison of these results shows that, although power-law scaling form emerges from the early times of the dynamics, there are noticeable differences, which do not produce negligible effects on the estimates of critical exponents. The observed universality of short-time critical behaviour supplements our fundamental knowledge of critical phenomena. However, short-time dynamics seem to be related only approximately with equilibrium critical properties. Therefore, any attempt to estimate equilibrium critical exponents from the short-time regime would be superfluous, except for the purpose of comparison, because these exponents have been determined very precisely by different methods.

Exact finite-size-scaling corrections to the critical two-dimensional Ising model on a torus: II. Triangular and hexagonal lattices

We compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulas for the first coefficients of each expansion. We analyze the implications of these findings on the renormalization-group description of the model.

Universal relations in the finite-size correction terms of two-dimensional Ising models.

Quite recently, Izmailian and Hu [Phys. Rev. Lett. 86, 5160 (2001)] studied the finite-size correction terms for the free energy per spin and the inverse correlation length of the critical two-dimensional Ising model. They obtained the universal amplitude ratio for the coefficients of two series. In this study we give a simple derivation of this universal relation; we do not use an explicit form of series expansion. Moreover, we show that the Izmailian and Hu's relation is reduced to a simple and exact relation between the free energy and the correlation length. This equation holds at any temperature and has the same form as the finite-size scaling.

Crossing probabilities on same-spin clusters in the two-dimensional Ising model

Probabilities of crossing on same-spin clusters, seen as order parameters, have been introduced recently for the critical two-dimensional Ising model by Langlands, Lewis and Saint-Aubin. We extend Cardy's ideas, introduced for percolation, to obtain an ordinary differential equation of order six for the horizontal crossing probability πh. Due to the identity πh(r) + πh(1/r) = 1, the function πh must lie in a three-dimensional subspace. New measurements of πh are made for 40 values of the aspect ratio r (r[0.1443,6.928]). These data are more precise than those obtained by Langlands et al as the 95%-confidence interval is brought to 4×10-4. A three-parameter fit using these new data determines the solution of the differential equation. The largest gap between this solution and the 40 data is smaller than 4×10-4. The probability πhv of simultaneous horizontal and vertical crossings is also treated.

Exact finite-size-scaling corrections to the critical two-dimensional Ising model on a torus

We analyse the finite-size corrections to the energy and specific heat of the critical two-dimensional spin-½ Ising model on a torus. We extend the analysis of Ferdinand and Fisher to compute the correction of order L-3 to the energy and the corrections of order L-2 and L-3 to the specific heat. We also obtain general results on the form of the finite-size corrections to these quantities: only integer powers of L-1 occur, unmodified by logarithms (except of course for the leading log L term in the specific heat); and the energy expansion contains only odd powers of L-1. In the specific-heat expansion any power of L-1 can appear, but the coefficients of the odd powers are proportional to the corresponding coefficients of the energy expansion.