Critical Ising Model Research Articles

On multiple SLE for the FK–Ising model

We prove the convergence of multiple interfaces in the critical planar q=2 random cluster model and provide an explicit description of the scaling limit. Remarkably, the expression for the partition function of the resulting multiple SLE16/3 coincides with the bulk spin correlation in the critical Ising model in the half-plane, after formally replacing a position of each spin and its complex conjugate with a pair of points on the real line. As a corollary we recover Belavin–Polyakov–Zamolodchikov equations for the spin correlations.

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).

Differentiable programming of isometric tensor networks

Differentiable programming is a new programming paradigm which enables large scale optimization through automatic calculation of gradients also known as auto-differentiation. This concept emerges from deep learning, and has also been generalized to tensor network optimizations. Here, we extend the differentiable programming to tensor networks with isometric constraints with applications to multiscale entanglement renormalization ansatz (MERA) and tensor network renormalization (TNR). By introducing several gradient-based optimization methods for the isometric tensor network and comparing with Evenbly–Vidal method, we show that auto-differentiation has a better performance for both stability and accuracy. We numerically tested our methods on 1D critical quantum Ising spin chain and 2D classical Ising model. We calculate the ground state energy for the 1D quantum model and internal energy for the classical model, and scaling dimensions of scaling operators and find they all agree with the theory well.

On the scaling behaviour and conformal properties of triangular Ising model with three-spin interactions at the critical point

The scaling behaviour and conformal properties of a triangular Ising model with three-spin interactions at the critical point are studied using extensive Monte Carlo simulations. We show that in the continuum limit the external perimeter of spin clusters are conformally invariant and statistically equivalent to Schramm–Loewner evolution (SLE κ ) curves with diffusivity κ = 4, indicating that the loop ensembles associated with the critical Ising model with three-spin interactions, belonging to the universality class of domain walls in the four-states Potts model. The left-passage probabilities, distribution of the winding angle, and the driving function distribution for the interfaces are consistent with each other. We also study many geometrical properties of the critical interfaces to characterize the fractal structure and the other scaling properties of conformal loop ensembles. To this end, the fractal dimension of contours, the scaling properties of length and areas distributions and Zipf’s laws, are demonstrated numerically. Our results are compatible with the hyper-scaling relation for spin models.

Constrained percolation in two dimensions

We prove absence of infinite clusters and contours in a class of critical constrained percolation models on the square lattice. The percolation configuration is assumed to satisfy certain hard local constraints, but only weak symmetry and ergodicity conditions are imposed on its law. The proofs use new combinatorial techniques exploiting planar duality. Applications include absence of infinite clusters of diagonal edges for critical dimer models on the square-octagon lattice, as well as absence of infinite contours and infinite clusters for critical XOR Ising models on the square grid. We also prove that there exists at most one infinite contour for high-temperature XOR Ising models, and no infinite contour for low-temperature XOR Ising model.

A formula for boundary correlations of the critical Ising model

Given a finite rhombus tiling of a polygonal region in the plane, the associated critical Z-invariant Ising model is invariant under star-triangle transformations. We give a simple matrix formula describing spin correlations between boundary vertices in terms of the shape of the region. When the region is a regular polygon, our formula becomes an explicit trigonometric sum.

Distributions in CFT. Part II. Minkowski space

CFTs in Euclidean signature satisfy well-accepted rules, such as the convergent Euclidean OPE. It is nowadays common to assume that CFT correlators exist and have various properties also in Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. Here we consider Wightman 4-point functions of scalar primaries in Lorentzian signature. We derive a minimal set of their properties solely from the Euclidean unitary CFT axioms, without using extra assumptions. We establish all Wightman axioms (temperedness, spectral property, local commutativity, clustering), Lorentzian conformal invariance, and distributional convergence of the s-channel Lorentzian OPE. This is done constructively, by analytically continuing the 4-point functions using the s-channel OPE expansion in the radial cross-ratios ρ, overline{rho} . We prove a key fact that |ρ|, left|overline{rho}right| < 1 inside the forward tube, and set bounds on how fast |ρ|, left|overline{rho}right| may tend to 1 when approaching the Minkowski space.We also provide a guide to the axiomatic QFT literature for the modern CFT audience. We review the Wightman and Osterwalder-Schrader (OS) axioms for Lorentzian and Euclidean QFTs, and the celebrated OS theorem connecting them. We also review a classic result of Mack about the distributional OPE convergence. Some of the classic arguments turn out useful in our setup. Others fall short of our needs due to Lorentzian assumptions (Mack) or unverifiable Euclidean assumptions (OS theorem).

The effect of free boundary conditions on the Ising model in high dimensions

We study the critical Ising model with free boundary conditions on finite domains in $${\mathbb {Z}}^d$$ with $$d\ge 4$$ . Under the assumption, so far only proved completely for high d, that the critical infinite volume two-point function is of order $$|x-y|^{-(d-2)}$$ for large $$|x-y|$$ , we prove the same is valid on large finite cubes with free boundary conditions, as long as x, y are not too close to the boundary. This confirms a numerical prediction in the physics literature by showing that the critical susceptibility in a finite domain of linear size L with free boundary conditions is of order $$L^2$$ as $$L\rightarrow \infty $$ . We also prove that the scaling limit of the near-critical (small external field) Ising magnetization field with free boundary conditions is Gaussian with the same covariance as the critical scaling limit, and thus the correlations do not decay exponentially. This is very different from the situation in low d or the expected behavior in high d with bulk boundary conditions.

Two-dimensional critical systems with mixed boundary conditions: Exact Ising results from conformal invariance and boundary-operator expansions.

With conformal-invariance methods, Burkhardt, Guim, and Xue studied the critical Ising model, defined on the upper half plane y>0 with different boundary conditions a and b on the negative and positive x axes. For ab=-+ and f+, they determined the one- and two-point averages of the spin σ and energy ε. Here +,-, and f stand for spin-up, spin-down, and free-spin boundaries, respectively. The case +-+-+⋯, where the boundary condition switches between + and - at arbitrary points, ζ_{1},ζ_{2},⋯ on the x axis was also analyzed. In the first half of this paper a similar study is carried out for the alternating boundary condition +f+f+⋯ and the case -f+ of three different boundary conditions. Exact results for the one- and two-point averages of σ,ε, and the stress tensor T are derived with conformal-invariance methods. From the results for 〈T〉, the critical Casimir interaction with the boundary of a wedge-shaped inclusion is derived for mixed boundary conditions. In the second half of the paper, arbitrary two-dimensional critical systems with mixed boundary conditions are analyzed with boundary-operator expansions. Two distinct types of expansions-away from switching points of the boundary condition and at switching points-are considered. Using the expansions, we express the asymptotic behavior of two-point averages near boundaries in terms of one-point averages. We also consider the strip geometry with mixed boundary conditions and derive the distant-wall corrections to one-point averages near one edge due to the other edge. Finally we confirm the consistency of the predictions obtained with conformal-invariance methods and with boundary-operator expansions, in the the first and second halves of the paper.

Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph.

The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q=2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q=2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q=1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q=1 instead of q=2. This demonstrates the percolation effects in the FK Ising model on the complete graph.

Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model

In this article we study imaginary Gaussian multiplicative chaos—namely a family of random generalized functions which can formally be written as $e^{iX(x)}$, where $X$ is a log-correlated real-valued Gaussian field on $\mathbf{R}^{d}$, that is, it has a logarithmic singularity on the diagonal of its covariance. We study basic analytic properties of these random generalized functions, such as what spaces of distributions these objects live in, along with their basic stochastic properties, such as moment and tail estimates. After this, we discuss connections between imaginary multiplicative chaos and the critical planar Ising model, namely that the scaling limit of the spin field of the critical planar XOR-Ising model can be expressed in terms of the cosine of the Gaussian free field, that is, the real part of an imaginary multiplicative chaos distribution. Moreover, if one adds a magnetic perturbation to the XOR-Ising model, then the scaling limit of the spin field can be expressed in terms of the cosine of the sine-Gordon field, which can also be viewed as the real part of an imaginary multiplicative chaos distribution. The first sections of the article have been written in the style of a review, and we hope that the text will also serve as an introduction to imaginary chaos for an uninitiated reader.

The critical two dimensional Ising model on right-triangle-shaped square lattices

Using the bond-propagation algorithm, we study the critical Ising model on the square lattice with the shape of right triangle and free boundaries. The length of short leg is N and that of long leg is M. The aspect ratio between the two legs ρ=M∕N is fixed. For six ratios: ρ=1,2,3,4,8,16, the critical free energy density, internal energy density and specific heat on the lattices with 30≤N≤600 are calculated. Based on these accurate data, we determine exact expansions of the critical free energy, internal energy, and specific heat. With these expansions, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat. The corner term in free energy density proportional to lnN is consistent to the conformal field theory in the accuracy of 10−10 at the least. It is found that the edge terms in the internal energy proportional to lnN related to the hypotenuse are the same for ρ=1,2,3,4,8,16, i.e. this term is geometry independent. However the edge terms in the specific heat proportional to lnN are geometry dependent, i.e. they are different for different ρ. The corner terms in the internal energy and specific heat are also obtained.

Finitary codings for spatial mixing Markov random fields

It has been shown by van den Berg and Steif (Ann. Probab. 27 (1999) 1501–1522) that the subcritical and critical Ising model on $\mathbb{Z}^{d}$ is a finitary factor of an i.i.d. process (ffiid), whereas the super-critical model is not. In fact, they showed that the latter is a general phenomenon in that a phase transition presents an obstruction for being ffiid. The question remained whether this is the only such obstruction. We make progress on this, showing that certain spatial mixing conditions (notions of weak dependence on boundary conditions, not to be confused with other notions of mixing in ergodic theory) imply ffiid. Our main result is that weak spatial mixing implies ffiid with power-law tails for the coding radius, and that strong spatial mixing implies ffiid with exponential tails for the coding radius. The weak spatial mixing condition can be relaxed to a condition which is satisfied by some critical two-dimensional models. Using a result of the author (Spinka (2018)), we deduce that strong spatial mixing also implies ffiid with stretched-exponential tails from a finite-valued i.i.d. process. We give several applications to models such as the Potts model, proper colorings, the hard-core model, the Widom–Rowlinson model and the beach model. For instance, for the ferromagnetic $q$-state Potts model on $\mathbb{Z}^{d}$ at inverse temperature $\beta $, we show that it is ffiid with exponential tails if $\beta $ is sufficiently small, it is ffiid if $\beta <\beta _{c}(q,d)$, it is not ffiid if $\beta >\beta_{c}(q,d)$ and, when $d=2$ and $\beta =\beta _{c}(q,d)$, it is ffiid if and only if $q\le 4$.

Gauging defects in quantum spin systems: A case study

The goal of this work is to build a dynamical theory of defects for quantum spin systems. A kinematic theory for an indefinite number of defects is first introduced exploiting distinguishable Fock space. Dynamics are then incorporated by allowing the defects to become mobile via a microscopic Hamiltonian. This construction is extended to topologically ordered systems by restricting to the ground state eigenspace of Hamiltonians generalizing the golden chain. We illustrate the construction with the example of a spin chain with $\mathbf{Vec}(\mathbb{Z}/2\mathbb{Z})$ fusion rules, employing generalized tube algebra techniques to model the defects in the chain. The resulting dynamical defect model is equivalent to the critical transverse Ising model.

Critical Casimir interaction between colloidal Janus-type particles in two spatial dimensions

We study colloidal particles with chemically inhomogeneous surfaces suspended in a critical binary liquid mixture. The inhomogeneous particle surface is composed of patches with alternating adsorption preferences for the two components of the binary solvent. By describing the binary liquid mixture at its consolute point in terms of the critical Ising model we exploit its conformal invariance in two spatial dimension. This allows us to determine exactly the universal profiles of the order parameter, the energy density, and the stress tensor as well as some of their correlation functions around a single particle for various shapes and configurations of the surface patches. The formalism encompasses several interesting configurations, including Janus particles of circular and needle shapes with dipolar symmetry and a circular particle with quadrupolar symmetry. From these single-particle properties we construct the so-called small particle operator expansion (SPOE), which enables us to obtain asymptotically exact expressions for the position- and orientation-dependent critical Casimir interactions of the particles with distant objects, such as another particle or the confining walls of a half plane, strip, or wedge, with various boundary conditions for the order parameter. In several cases we compare the interactions at large distances with the ones at close distance (but still large on the molecular scale). We also compare our analytical results for two Janus particles with recent simulation data.

Critical Ising Model on Random Triangulations of the Disk: Enumeration and Local Limits

We consider Boltzmann random triangulations coupled to the Ising model on their faces, under Dobrushin boundary conditions and at the critical point of the model. The first part of this paper computes explicitly the partition function of this model by solving its Tutte’s equation, extending a previous result by Bernardi and Bousquet-Mélou (J Combin Theory Ser B 101(5):315–377, 2011) to the model with Dobrushin boundary conditions. We show that the perimeter exponent of the model is 7/3 in contrast to the exponent 5/2 for uniform triangulations. In the second part, we show that the model has a local limit in distribution when the two components of the Dobrushin boundary tend to infinity one after the other. The local limit is constructed explicitly using the peeling process along an Ising interface. Moreover, we show that the main interface in the local limit touches the (infinite) boundary almost surely only finitely many times, a behavior opposite to that of the Bernoulli percolation on uniform maps. Some scaling limits closely related to the perimeters of finite clusters are also obtained.

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 $N\rightarrow\infty$.

Scaling Hypothesis for Matrix Product States.

We study critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. The critical point, exponents, and central charge are determined by optimizing them to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling Ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of λϕ^{4} with mass parameter μ^{2} and lattice spacing a, we demonstrate a double data collapse for the correlation length δξ(μ,λ,D)=ξ[over ˜]((α-α_{c})(δ/a)^{-1/ν}) with D the bond dimension, δ the gap between eigenvalues of the transfer matrix, and α_{c}=μ_{R}^{2}/λ the parameter which fixes the critical quantum field theory.

Non-self-averaging in the Critical Point of a Random Ising Ferromagnet

In this paper, we review recent results on sample-to-sample fluctuations in a critical Ising model with quenched random ferromagnetic couplings. In particular, in terms of the renormalized replica Ginzburg–Landau Hamiltonian in dimensions D < 4 an explicit expression for the probability distribution function (PDF) of the critical free energy fluctuations is derived. Next, using known fixed-point values for the renormalized coupling parameters the universal curve for such PDF in the dimension D = 3 is obtained. For the specific case of the two-dimensional Ising model, using replica calculations in the renormalization group framework, we derive explicit expressions for the PDF of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energy is Gaussian, and its typical sample-to-sample fluctuations as well as its 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 → ∞.

Circle Patterns and Critical Ising Models

A circle pattern is an embedding of a planar graph in which each face is inscribed in a circle. We define and prove magnetic criticality of a large family of Ising models on planar graphs whose dual is a circle pattern. Our construction includes as a special case the critical isoradial Ising models of Baxter.