# Connecting Complex Electronic Pattern Formation to Critical Exponents

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

_{2}and NdNiO

_{3}as they transition from metal to insulator, via scanning near-field optical microscopy [7,8,26] and via scanning X-ray microscopy [9]. Using the cluster analysis techniques we developed, we showed that in these cases the dominant type of disorder driving the pattern formation is in the random field [27] universality class, and we argued that this is the origin of nonequilibrium behavior in both systems [8,9,14,15,16].

_{2}. For the two examples above, our cluster analysis [14,15,16] of the image data shows that the geometric clusters (which are defined as the connected set of the nearest neighbor sites with Ising variables being the same value) extracted from the multiscale pattern formation display universal scaling behavior over multiple decades, suggesting criticality and universality as the origin of the spatial complexity revealed by scanning probe microscopy in strongly correlated electronic systems. By comparing the data-extracted critical exponents derived from the self-similarity of the geometric clusters with the theoretical values for the fixed points contained in Equation (1), we have shown [14] that it is possible to identify the universality class governing the scaling behavior of the geometric clusters observed on surfaces of novel materials.

## 3. Where Is the Connectivity Function Power Law?

**Conjecture**

**#1:**

## 4. Geometric Criticality and Thermodynamic Criticality: Two Types of Critical Exponents

**Conjecture**

**#2:**

## 5. Numerical Results for the Critical Exponents Defined on 2D Slices of a 3D System

## 6. Numerical Results for the Connectivity Function in the Clean 2D Ising Model

## 7. Relation of Infinite Cluster to Bulk Magnetization on a 2D Slice

## 8. Two Correlation Lengths

## 9. Discussion

## 10. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Geometric Clusters and Fortuin-Kasteleyn Clusters. (

**a**) An example of an Ising configuration. Blue circles represent up spins; red circles represent down spins. Geometric clusters are nearest-neighbor connected sets of like spins. There are two red geometric clusters, with connecting bonds denoted in green. The blue sites also form a single geometric cluster. (

**b**) To construct the FK clusters, allow the bonds of each geometric cluster to be broken with a Boltzmann-like probability, to mimic temperature fluctuations [18]. (

**c**) The remaining set of nearest-neighbor connected clusters are the FK clusters.

**Figure 2.**Cluster size distribution $D\left(s\right)\sim {s}^{-\tau}$ of internal C-3Dx geometric clusters at ${T}_{p}^{\mathrm{slice}}={T}_{c}^{3D}$ for system sizes from $L=80$ to $L=192$. The inset (

**a**) shows the extrapolation of $\tau $ from the DLD fits in the main panel to the thermodynamic limit $L\to \infty $, which gives $\tau =2.001\pm 0.013$. The inset (

**b**) shows the scaling collapse of the curves in the main panel, using the extrapolated $\tau $ in (

**a**).

**Figure 3.**The power law fits for ${R}_{s}\sim {s}^{1/{d}_{v}}$, using internal C-3Dx geometric clusters at ${T}_{p}^{\mathrm{slice}}={T}_{c}^{3D}$ for system sizes from $L=80$ to $L=192$. The inset (

**a**) shows the extrapolation of ${d}_{v}$ from the fits in the main panel to the thermodynamic limit $L\to \infty $, which gives ${d}_{v}=1.856\pm 0.018$. The inset (

**b**) shows the scaling collapse of curves in the main panel, using the extrapolated ${d}_{v}$ in (

**a**).

**Figure 4.**The power law fits for ${R}_{h}\sim {h}^{1/{d}_{h}}$, using internal C-3Dx geometric clusters at ${T}_{p}^{\mathrm{slice}}={T}_{c}^{3D}$ for system sizes from $L=80$ to $L=192$. The inset (

**a**) shows the extrapolation of ${d}_{h}$ from the fits in the main panel to the thermodynamic limit $L\to \infty $, which gives ${d}_{h}=1.714\pm 0.022$. The inset (

**b**) shows the scaling collapse of curves in the main panel, using the extrapolated ${d}_{h}$ in (

**a**).

**Figure 5.**Pair connectivity function ${g}_{\mathrm{conn}}\left(r\right)$ on the 2D slice of clean 3D Ising system with different system sizes from $L=80$ to $L=192$ at ${T}_{p}^{\mathrm{slice}}={T}_{c}^{3D}$. The inset (

**a**) shows the extrapolation of $d-2+{\eta}_{p}$ from the scaling form fits of ${g}_{\mathrm{conn}}\left(r\right)$ in the main panel to the thermodynamic limit $L\to \infty $, which yields $d-2+{\eta}_{p}=0.322\pm 0.002$. The inset (

**b**) shows the scaling collapse of curves in the main panel, using the extrapolated $d-2+{\eta}_{p}$ in (

**a**).

**Figure 6.**Pair connectivity function ${g}_{\mathrm{conn}}\left(r\right)$ on the 2D slice of the clean 3D Ising system with size $L=192$ at different temperatures around ${T}_{p}^{\mathrm{slice}}={T}_{c}^{3D}$. ${g}_{\mathrm{conn}}\left(r\right)$ turns up at large r for $T<{T}_{c}^{3D}$.

**Figure 7.**Pair connectivity function ${g}_{\mathrm{conn}}\left(r\right)$ of the 2D clean Ising system with different sizes from $L=96$ to $L=352$ at ${T}_{p}^{2D}={T}_{c}^{2D}$. The inset (

**a**) shows the extrapolation of $d-2+{\eta}_{p}$ from the scaling form fits of ${g}_{\mathrm{conn}}\left(r\right)$ in the main panel to the thermodynamic limit $L\to \infty $, which yields $d-2+{\eta}_{p}=0.104\pm 0.002$. The inset (

**b**) shows the scaling collapse of curves in the main panel, using the extrapolated $d-2+{\eta}_{p}$ in (

**a**).

**Figure 8.**Pair connectivity function ${g}_{\mathrm{conn}}\left(r\right)$ of the 2D clean Ising system with size $L=256$ at different temperatures around ${T}_{p}^{2D}={T}_{c}^{2D}$. The change of behavior for ${g}_{\mathrm{conn}}\left(r\right)$ from the scaling form shape to the turning-up shape happens in the close vicinity of ${T}_{p}^{2D}={T}_{c}^{2D}$.

**Figure 9.**Illustration of the equilibrium 2D slice configuration embedded in the 3D Ising system with L = 192 at (

**a**) $T={T}_{p}^{3D}$ and (

**b**) $T={T}_{c}^{3D}$.

**Figure 10.**Numerical results of the reduced spontaneous magnetization M and the infinite network strength R of the 2D slice embedded in 3D Ising system under zero external field with $L=192$ at $T<{T}_{c}^{3D}$, averaged over a number of configurations in the order of magnitude ∼10${}^{5}$. M and R both approach 1 when T decreases, with $M<R$. The inset shows a log–log plot of M vs. R, compared to the power law scaling form ansatz $M\propto {R}^{2.37}$.

**Figure 11.**Pair connectivity function ${g}_{\mathrm{conn}}\left(r\right)$ and spin-spin correlation function ${g}_{\mathrm{spin}}\left(r\right)$ for (

**a**) C-3Dx with $L=160$ at the critical temperature ${T}_{c}^{3D}(L\to \infty )$ and (

**b**) C-2D with $L=256$ at the critical temperature ${T}_{c}^{2D}(L\to \infty )$. The black cross markers on the correlation functions denote the length scales of correlation lengths, extracted from a fit to Equations (4) and (11). The thick lines represent the power law behaviors of the correlation functions by setting the exponential terms to unity.

**Figure 12.**The two correlation lengths ${\xi}_{p}$ (pertaining to ${g}_{\mathrm{conn}}\left(r\right)$) and ${\xi}_{c}$ (pertaining to ${g}_{\mathrm{spin}}\left(r\right)$) for C-3Dx with $L=160$ for $T\ge {T}_{c}^{3D}(L\to \infty )$, computed by using the scaling form fits of the correlation functions, Equations (4) and (11). The dotted line marks the thermodynamic critical temperature in the thermodynamic limit.

Symbol | Definition |
---|---|

${\sigma}_{i}$ | Ising variable on a site i (Equation (1)) |

${J}^{\left|\right|}$; ${J}^{\perp}$ | In-plane and inter-plane Ising coupling constant (Equation (1)) |

h | Applied (uniform) magnetic field (Equation (1)) |

${h}_{i}$ | Random (local) field (Equation (1)) |

s | Number of sites in a nearest-neighbor connected cluster |

h | Number of sites on the hull of a cluster |

${R}_{s}$ | Radius of Gyration of a cluster (Equation (2)) |

${R}_{h}$ | Radius of Gyration of the interior of a cluster (Equation (3)) |

$D\left(s\right)$ | Distribution of cluster sizes |

${g}_{\mathrm{conn}}\left(r\right)$ | Pair connectivity function (Equation (4)) |

${g}_{\mathrm{spin}}\left(r\right)$ | Spin-spin correlation function (Equation (10)) |

${\xi}_{c}$ | Spin-spin correlation length (Equation (11) |

${\xi}_{p}$ | Percolation correlation length (Equation (4)) |

2D | Two-dimensional |

3D | Three-dimensional |

C-2D | Clean 2D Ising model (i.e., no disorder) |

C-3D | Clean 3D Ising model |

C-3Dx | Interior slice of C-3D |

T | Temperature |

${T}_{c}$ | Magnetic ordering temperature |

${T}_{p}$ | Percolaton temperature |

${T}_{p}^{\mathrm{slice}}$ | Percolation temperature on a slice |

${T}_{c}^{3D}$; ${T}_{p}^{3D}$ | Bulk (3D) ${T}_{c}$ and ${T}_{p}$ |

${T}_{c}^{3D}(L\to \infty )=4.51152786J$ | ${T}_{c}$ of C-3D in large system limit [19,20] |

${T}_{c}^{2D}(L\to \infty )=2/\mathrm{ln}(1+\sqrt{2})J$ | ${T}_{c}$ of C-2D for infinite size system [21] |

p | Percolation fraction |

${p}_{c}$ | Critical fraction for percolation |

FK | Fortuin–Kasteleyn |

L | Length of one side of simulation |

d | Spatial dimension |

R | Infinite network strength (Equation (5)) |

${R}_{\uparrow}^{slice}$; ${R}_{\downarrow}^{slice}$ | Ininite network strength of up (down) spins on a slice (Equation (5)) |

${\pi}_{i}$; ${\tilde{\pi}}_{i}$ | Lattice gas variables (Equation (6)) |

${\tilde{\gamma}}_{i}^{\infty}$ | Characteristic function that spin i belongs to the infinite 2D slice $(+)$-cluster |

${\gamma}_{i}^{\infty}$ | Characteristic function that spin i belongs to the infinite 2D slice $(-)$-cluster |

Name | Symbol | Definition |
---|---|---|

Fisher Exponent | $\tau $ | $D\left(s\right)\propto {s}^{\tau}$ |

Volume Fractal Dimension | ${d}_{v}$ | $s\propto {R}_{s}^{{d}_{v}}$ |

Hull Fractal Dimension | ${d}_{h}$ | $h\propto {R}_{h}^{{d}_{h}}$ |

$\beta $ | $M\sim |T-{T}_{c}{|}^{\beta}$ | |

$\nu $ | $\xi \sim |T-{T}_{c}{|}^{-\nu}$ | |

Anomalous Exponent for Magnetism (Equation (10) and (11)) | ${\eta}_{c}$ | ${g}_{\mathrm{spin}}\left(r\right)\sim {r}^{-(d-2+{\eta}_{c})}$ |

Anomalous Exponent for Percolation (Equation (4)) | ${\eta}_{p}$ | ${g}_{\mathrm{conn}}\left(r\right)\sim {r}^{-(d-2+{\eta}_{p})}$ |

${\eta}_{c}$ and ${\eta}_{p}$ as measured on a 2D (interior) slice | ${\eta}_{c}^{\left|\right|}$; ${\eta}_{p}^{\left|\right|}$ |

**Table 3.**Theoretical/numerical values for the 2D geometric cluster self-similarity-characterized critical percolation exponents for Ising models and standard percolation model.

Model | 2D Ising Model [33,48] | 2D Slice of 3D Ising Model (This Work) | Standard Percolation [47,49,50] |
---|---|---|---|

Fixed Point | C-2D | C-3Dx | P-2D |

$\tau $ | $379/187=2.027$ | $2.001\pm 0.013$ | $187/91=2.055$ |

${d}_{v}$ | $187/96=1.948$ | $1.856\pm 0.018$ | $91/48=1.896$ |

${d}_{h}$ | $11/8=1.375$ | $1.714\pm 0.022$ | $7/4=1.75$ |

$d-2+{\eta}_{p}$ | $0.104\pm 0.002$ (this work) | $0.322\pm 0.002$ | $5/24=0.208$ |

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Liu, S.; Carlson, E.W.; Dahmen, K.A.
Connecting Complex Electronic Pattern Formation to Critical Exponents. *Condens. Matter* **2021**, *6*, 39.
https://doi.org/10.3390/condmat6040039

**AMA Style**

Liu S, Carlson EW, Dahmen KA.
Connecting Complex Electronic Pattern Formation to Critical Exponents. *Condensed Matter*. 2021; 6(4):39.
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Liu, Shuo, Erica W. Carlson, and Karin A. Dahmen.
2021. "Connecting Complex Electronic Pattern Formation to Critical Exponents" *Condensed Matter* 6, no. 4: 39.
https://doi.org/10.3390/condmat6040039