Dynamic Phase Transition in 2D Ising Systems: Effect of Anisotropy and Defects
<p>Time−dependent average magnetization (orange) in presence of an oscillating field (blue) for different choices of temperature for an isotropic, defect-less Ising system. <b>Left</b>: The magnetization follows the field, albeit with a time delay. The system is in its dynamically disordered phase (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>1.9</mn> <mspace width="0.166667em"/> <mi>J</mi> </mrow> </semantics></math>). <b>Right</b>: The magnetization does not reverse its sign, and the system is in its dynamically ordered phase (here, <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>1.5</mn> <mspace width="0.166667em"/> <mi>J</mi> </mrow> </semantics></math>). Simulations over a 2D square system with size <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math>, averaging 20 cycles for a field with <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mspace width="0.166667em"/> <mi>J</mi> </mrow> </semantics></math> and period <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>258</mn> </mrow> </semantics></math>. For these values of the parameters, the dynamic critical temperature is <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Θ</mi> <mi mathvariant="normal">c</mi> </msub> <mo>=</mo> <mn>1.8</mn> <mspace width="0.166667em"/> <mi>J</mi> </mrow> </semantics></math>. The reported magnetic field <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> is measured in units of <span class="html-italic">J</span>.</p> "> Figure 2
<p>DPT in an anisotropic Ising system. <b>Top left</b>: Average magnetization per cycle as a function of temperature for heating/cooling experiments (no hysteresis effects). The external field amplitude is set to <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, the values chosen for the anisotropy <math display="inline"><semantics> <mi>λ</mi> </semantics></math> are displayed in the inset, and the points’ shapes correspond to different system sizes: <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> (*), 100 (+), 140 (∘), 180 (△), and 220 (◁). Finite-size effects are not relevant, as all points collapse into the same curves. <b>Top right</b>: Susceptibility curve as a function of temperature. The peak spots the critical temperature for the DPT. <b>Bottom</b>: Binder cumulant as a function of temperature for a fixed anisotropy <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>16</mn> <mi>π</mi> </mrow> </semantics></math>. The inset shows the separation of the curves for systems with different sizes in the vicinity of the critical temperature.</p> "> Figure 3
<p>Dynamic critical temperature as a function of the inverse system size for the anisotropic Ising model. For each choice of anisotropy, a solid line is shown to guide the eye. For <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>≳</mo> <mn>60</mn> </mrow> </semantics></math>, no significant variation in the critical temperature is seen.</p> "> Figure 4
<p>Critical temperature <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Θ</mi> <mi mathvariant="normal">c</mi> </msub> </semantics></math> for the DPT for a 2D anisotropic Ising model, as a function of the anisotropy coefficient <math display="inline"><semantics> <mi>λ</mi> </semantics></math>. The black dotted line represents the static critical temperature as derived in [<a href="#B38-entropy-26-00120" class="html-bibr">38</a>]. For each value of <math display="inline"><semantics> <msub> <mi>h</mi> <mn>0</mn> </msub> </semantics></math>, the factor <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>(</mo> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </semantics></math>, introduced in (<a href="#FD16-entropy-26-00120" class="html-disp-formula">16</a>) and reported in the right panel, has been computed by minimizing the residual error <math display="inline"><semantics> <mrow> <msub> <mo>∑</mo> <mi>λ</mi> </msub> <msup> <mfenced separators="" open="(" close=")"> <msub> <mi mathvariant="sans-serif">Θ</mi> <mi mathvariant="normal">c</mi> </msub> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>T</mi> <mi mathvariant="normal">c</mi> </msub> <mrow> <mo>(</mo> <mi>λ</mi> <mo>)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </mrow> </semantics></math>. The continuous plot corresponds to the thermodynamical critical temperature, computed from the analytical solution obtained in the absence of an oscillating field.</p> "> Figure 5
<p><b>Top Left</b>: Average magnetization per cycle as a function of temperature for a fixed external field amplitude <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, with three choices of the fraction of defects <span class="html-italic">f</span> and different system sizes: <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> (*), 100 (+), 180 (∘), 260 (△). <b>Top Right</b>: Dynamic susceptibility as a function of temperature. Their peaks locate the dynamic critical temperature. <b>Bottom</b>: Binder cumulant as a function of temperature for the system with defects <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>1</mn> <mo>%</mo> </mrow> </semantics></math>. The inset shows a zoomed-in region of the intersection of the curves.</p> "> Figure 6
<p>Dynamic critical temperature as a function of the inverse system size. Different choices of the fraction of defects are displayed for the case <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>. The dynamic critical temperature is determined in the thermodynamic limit by using a linear interpolation, in accordance with [<a href="#B24-entropy-26-00120" class="html-bibr">24</a>].</p> "> Figure 7
<p><b>Left</b>: Dynamical critical temperature as a function of the defect fraction. The linear extrapolation for <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math> in the <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> case coincides with the defect-free critical temperature derived in [<a href="#B24-entropy-26-00120" class="html-bibr">24</a>]. <b>Right</b>: Dynamical critical temperature as a function of the external field amplitude. The linear extrapolation for <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math> is compatible with the Curie temperature for all the data series.</p> "> Figure 8
<p>Correlation between the potential <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> and the local average magnetization per cycle <math display="inline"><semantics> <msub> <mi>Q</mi> <mi>i</mi> </msub> </semantics></math> at the dynamic critical temperature for a system with <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>260</mn> </mrow> </semantics></math>. The solid line represents the average value, while dashed lines delimit the standard deviation of the potential distribution of sites exhibiting the same value of <math display="inline"><semantics> <msub> <mi>Q</mi> <mi>i</mi> </msub> </semantics></math>.</p> "> Figure 9
<p>Sequence of defect configurations with increasing values of the potential index for a system with <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> and a defect fraction <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>2</mn> <mo>%</mo> </mrow> </semantics></math>. In the top row (from left to right): <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>def</mi> </msub> <mo>=</mo> <mn>0.049</mn> </mrow> </semantics></math>, 0.065, and 0.13. In the bottom row (again, from left to right): <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>def</mi> </msub> <mo>=</mo> <mn>0.26</mn> </mrow> </semantics></math>, 0.52, and 0.98. The color code of the plots is determined by the local potential <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math>. The color scale has been adjusted across the images; otherwise, the upper configurations would have all appeared predominantly orange. Yellow (black) squares represent positive (negative) defects.</p> "> Figure 10
<p>Log–log plot of the average potential index <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mi>def</mi> </msub> </semantics></math> (<b>left</b>) and its standard deviation <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>ϕ</mi> </msub> </semantics></math> (<b>right</b>), as a function of both the system size <span class="html-italic">L</span> and the defect fraction <span class="html-italic">f</span>. The two insets show that data associated with different values of <span class="html-italic">f</span> collapse into a universal curve by plotting <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi>def</mi> </msub> <msup> <mi>f</mi> <mrow> <mn>0.3</mn> </mrow> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>ϕ</mi> </msub> <msup> <mi>f</mi> <mrow> <mn>0.3</mn> </mrow> </msup> </mrow> </semantics></math>.</p> "> Figure 11
<p>Graphical representation of the Delaunay triangulation for the same realizations of defects shown in <a href="#entropy-26-00120-f009" class="html-fig">Figure 9</a>. We obtain, respectively, the values for the configuration index: <math display="inline"><semantics> <mrow> <mi mathvariant="script">A</mi> <mo>=</mo> <mn>1.42</mn> </mrow> </semantics></math>, 1.38, and 1.46 for the top row, and <math display="inline"><semantics> <mrow> <mi mathvariant="script">A</mi> <mo>=</mo> <mn>1.76</mn> </mrow> </semantics></math>, 2.17, and 1.46 for the bottom row. The colors specify triangles with different values of <math display="inline"><semantics> <msub> <mi>e</mi> <mi>i</mi> </msub> </semantics></math>: dark blue for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mrow> </semantics></math>, light blue for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>, orange for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and red for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>.</p> "> Figure 12
<p>Correlations between dynamic critical temperature (<b>left column</b>) and dynamic susceptibility peak (<b>right column</b>) and two geometric parameters: the total potential (<b>top row</b>) and the normalized area index (<b>bottom row</b>). For the total potential, each point represents the average over 10 realizations of defects with the specified target potential. The error bar represents the standard deviation. For the normalized area index, all 200 realizations have been analyzed, with a data binning procedure of 18 bins. The point represents the average dynamical property for each bin, and the error bar is its standard deviation. In all the panels, the orange window locates the interval of values where most random configurations can be found, as the position and width of the window are determined by the average and standard deviation of the geometric quantity as computed among random configurations. The thermodynamic properties were obtained by following 20 systems across <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>5000</mn> </mrow> </semantics></math> cycles under an oscillating magnetic field with intensity <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> for a system of size <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
2. Model
2.1. Time Evolution
2.2. Anisotropic Ising Model
2.3. Magnetic System with Defects
- A potential index , emerging from interpreting the defect configuration as a charge distribution;
- The dipole moment d of the defect configuration;
- A normalized area index , extracted from the Delaunay triangulation associated with the defect network.
2.3.1. Potential Index
2.3.2. Dipole Moment
2.3.3. Area Index
3. Dynamic Phase Transition
3.1. DPT in Anisotropic Magnetic Systems
3.2. DPT in Magnetic Systems with Defects
4. Understanding the Dynamic Behavior of Random Systems
4.1. Defect Potential
4.2. A Priori Estimates of the Dynamic Response
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Ettori, F.; Coupé, T.; Sluckin, T.J.; Puppin, E.; Biscari, P. Dynamic Phase Transition in 2D Ising Systems: Effect of Anisotropy and Defects. Entropy 2024, 26, 120. https://doi.org/10.3390/e26020120
Ettori F, Coupé T, Sluckin TJ, Puppin E, Biscari P. Dynamic Phase Transition in 2D Ising Systems: Effect of Anisotropy and Defects. Entropy. 2024; 26(2):120. https://doi.org/10.3390/e26020120
Chicago/Turabian StyleEttori, Federico, Thibaud Coupé, Timothy J. Sluckin, Ezio Puppin, and Paolo Biscari. 2024. "Dynamic Phase Transition in 2D Ising Systems: Effect of Anisotropy and Defects" Entropy 26, no. 2: 120. https://doi.org/10.3390/e26020120