# From Spin Glasses to Negative-Weight Percolation

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## Abstract

**:**

## 1. Spin Glasses, Ground States and Domain Walls

## 2. Graphs

**Definition**

**1.**

**Example**

**1.**

**Example**

**2.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Example**

**3.**

**Definition**

**5.**

## 3. Minimum-Weight Paths

## 4. Negative-Weight Percolation

- a phase with “small” lattice loops. This means that the lengths of the loops are short as compared to the size of the system; see Figure 10a (therein, the loop “size” is taken as the projection on the lattice axes). This resembles a diluted gas of loops.
- another phase exhibiting “large” loops resulting in dense configurations of packed loops. The entire lattice is eventually spanned, i.e., the loops “percolate”; see Figure 10c.

#### 4.1. Problem Definition of the NWP Problem and Algorithms

- bimodal$\pm J$ edge-weight distributions (usually $J=1$) [62]$$P(\omega )=\rho \delta (\omega +J)+(1-\rho )\delta (\omega -J)\phantom{\rule{0.166667em}{0ex}}.$$The fraction of negative weights is denoted by $\rho $.
- Weights can either exhibit a unit weight (with probability 1 − $\rho $), or, with probability $\rho $, they are drawn according to a Gaussian distribution (zero mean and variance one). This is described by the following probability density:$$P(\omega )=\rho exp(-{\omega}^{2}/2)/\sqrt{2\pi}+(1-\rho )\delta (\omega -1)\phantom{\rule{0.166667em}{0ex}}.$$This distribution is called mixed Gaussian here.

- An auxiliary graph ${G}_{\mathrm{A}}$ is created in two steps. the edges are treated first and then the nodes of the original graphs: Each edge, which joins adjacent sites within the original graph G, is replaced by a path made by three edges. For this purpose, one has to introduce two “additional” sites for each edge of the original graph. One of the two new edges which connects an original site to an additional site gets assigned a weight with the same value as the corresponding edge in the original graph. The other two edges of such a three-edge path get assigned a weight of zero.Next, each original site $i\in V$ is “duplicated”. This means i is replaced by two nodes ${i}_{1},{i}_{2}$. Correspondingly, all their incident edges and their weights are duplicated as well. Furthermore, one additional edge $\{{i}_{1},{i}_{2}\}$ with zero weight is created to link the duplicated sites ${i}_{1}$ and ${i}_{2}$, for each pair ${i}_{1},{i}_{2}$. These edges have weight zero. The result for the sample auxiliary graph ${G}_{\mathrm{A}}=({V}_{\mathrm{A}},{E}_{\mathrm{A}})$ is displayed in Figure 11II. Here, diamonds are used to depict the additional sites and circles to depict the duplicated sites. The weights assigned in the transformed graph ${G}_{\mathrm{A}}$ are also included in Figure 11II. A more extensive description of the mapping can be found in the literature [30,69].
- Next, for the auxiliary graph, an exact MWPM is obtained by applying combinatorial-optimisation algorithms [70,72]. This matching is given by a subset $M\subset {E}_{\mathrm{A}}$ with minimum weight, such that each node from the node set ${V}_{\mathrm{A}}$ is incident to exactly one edge of M. An illustration is shown in Figure 11III. Here, solid edges display M for the chosen assignment of weights. The dashed edges are not part of the matching.
- Finally, one basically inverts the transformation (1), which allows one to obtain a configuration of negative-weighted loops $\mathcal{C}$ on G from the matched edges M of the auxiliary graph. To understand this, note that each matching edge which is incident to a duplicated site (circle) and an additional site (square) will contribute to a loop; see Figure 11IV. More precisely, for each loop segment on G, there are always two such edges in M. Every matching edge which connects sites of the same type (i.e., additional with additional or duplicated with duplicated) carries a weight of zero and is not contained in a loop concerning G.After the set of loop edges is constructed, one can use a depth-first search [16,69] to actually find the set $\mathcal{C}$ of loops. For analysis, the geometric properties of all loops can be calculated. In the example shown in Figure 11I, there exists only one loop with negative weight with ${\omega}_{\mathcal{L}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}-2$. It has the length $\ell =8$.

- An additional node in ${G}_{\mathrm{A}}$ is matched with a duplicated node (see Figure 12I). In this case, the other duplicated node must be matched to an additional node as well because his “twin” is not available any more. This exactly models the path property: a path which enters a node (modelled by a pair of duplicated nodes in ${G}_{\mathrm{A}}$), corresponding to an additional-duplicated match, must leave the node again, i.e., there must be a second additional-duplicated match. For this reason, the weights are attached to the edges which connect additional to duplicated nodes.
- An additional node from ${G}_{\mathrm{A}}$ is matched to a second additional node. This represents the case that a path segment is absent here. This is why we assign the weight 0 for edges connecting additional to additional nodes. Such a match does not influence how the duplicated nodes are matched (see Figure 12II). It is possible that a node of G does not participate in a path or loop, in which case all adjacent additional nodes in ${G}_{\mathrm{A}}$ are matched to additional nodes (see Figure 12III). In this case, to obtain a perfect matching, the two duplicated nodes must be matched to each other.

#### 4.2. Paths

#### 4.3. Other Graphs

#### 4.4. Intersecting Loops or Paths

#### 4.5. Directed NWP

## 5. Results for NWP

#### 5.1. Percolation Transitions

`autoScale.py Python`tool [79]. In this case, the data collapse yielded the best quality for values $\rho =0.340(1)$ and $\nu =1.49(7)$. The quality of the collapse, i.e., the mean deviation of the data points of the rescaled curves, normalised by the error bars, was $S=0.91$ which is very good.

#### 5.2. Mean Field Behaviour of the NWP Model

#### 5.3. Phase Transitions in Diluted Negative Weight Percolation Models

- For type I dilution, the disorder is realised by selecting a fraction ${p}_{\mathrm{I}}$ of edges where the weights ${\omega}_{ij}$ are zero. Thus, the paths or loops can use these edges without changing the weight.
- For type II dilution, a random fraction ${p}_{\mathrm{II}}$ of edges is actually absent. Thus, no paths or loops can run there.

#### 5.4. $2D$ NWP and SLE

#### 5.5. Directed NWP

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FSS | finite-size scaling |

GS | ground state |

NWP | negative weight percolation |

RKKY | Ruderman, Kittel, Kasuya, Yosida |

MWP | minimum-weight path |

MWPM | minimum-weight perfect matching |

SAW | self avoiding walk |

SLE | Schramm–Loewner evolution |

SLPF | Schramm’s left-passage formula |

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**Figure 1.**Energy of two spins placed at distance r coupled through a cloud of conducting electrons, yielding the RKKY (Ruderman, Kittel, Kasuya, Yosida) interaction.

**Figure 2.**The ground state of a frustrated system consisting of four spins. Ferromagnetic interactions (“bonds”) are represented by straight lines. Thus, the interactions favour parallel orientation of the spins. An antiferromagnetic interaction is shown as a zigzag line. Independent of the orientation spin 2, one of its incident bonds is not satisfied. On the other hand, bonds 3-4 and 1-4 are satisfied.

**Figure 3.**Example for a realisation of a two-dimensional spin glass (free boundary conditions). The Spins are located on the sites of a square lattice. Nearest neighbour spins interact either through a ferromagnetic (straight blue line) or antiferromagnetic (jagged red line) bond. For this example, a bimodal bond distribution is assumed, which means all bonds exhibit $|{J}_{ij}|=J$.

**Figure 4.**The same two-dimensional realisation as shown in Figure 3. Spins are located on the sites denoted by open circles. Filled circle symbols denote the sites of the dual lattice. Cycles on the dual lattice correspond to closed domain walls in the original lattice. Those cycles which exhibit a positive energy are indicated by dashed lines plus a grey colour for the enclosed areas.

**Figure 5.**The same two-dimensional realisation as shown in Figure 3. Spins are located on the sites denoted by open circles. Filled circles denote the sites of the dual lattice. A ground state is depicted, where spins pointing downwards are shown, all other spins point upwards. The bonds which are not satisfied are marked grey. There does not exist a closed domain wall with positive energy.

**Figure 6.**

**Top**: An additional column of very strong bonds (shown at the right with very thick lines) is added for the example spin glass as shown in Figure 3. The bonds are not compatible with the GS configuration. This will force the spins in the first and last column to flip relative to each other and force a domain wall of minimum energy (dashed line) into the system.

**Bottom**: New ground state for the modified system. The spins on the left inside the marked area have been flipped. In both cases, again, spins are located on the sites denoted by open circles. Filled circles denote the sites of the dual lattice. The bonds which are not satisfied are marked grey.

**Figure 9.**Illustration of matching for a sample graph of six nodes and four edges. In the left, the matching consist of two edges $\{1,4\}$ and $\{2,5\}$. Matched edges and matched nodes are shown in bold lines. On the right, a perfect matching is shown, i.e., all nodes are matched.

**Figure 10.**Illustration of the NWP phase transition. Sample configurations of loops on a square grid for $L\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}96$ side length, with periodic boundary conditions. (Non) percolating loops are shown in black (grey). The configurations are taken for different values of the parameter $\rho $ which controls the disorder. (

**a**) $\rho <{\rho}_{c}$, (

**b**) $\rho \approx {\rho}_{c}$, and, (

**c**) $\rho >{\rho}_{c}$. Beyond the critical point ${\rho}_{c}$, in the thermodynamic limit $L\to \infty $, there exist paths which span the lattice in any direction which exhibits periodic boundaries.

**Figure 11.**Examples of the main steps of the algorithmic procedure: (

**I**) the original lattice G together with weights of edges; (

**II**) the auxiliary graph ${G}_{\mathrm{A}}$ with corresponding assignment of weights. Black edges exhibit the same value of the weight as the corresponding edge in the original graph and grey edges exhibit a weight of zero. Diamond symbols are used to mark the “additional” sites; (

**III**) minimum-weight perfect matching (MWPM) M: the matched edges are shown in bold and unmatched edges using dashed lines, and (

**IV**) loop configuration (bold edges) that is the result of the MWPM shown in (

**III**).

**Figure 12.**How the matching models detects loops: (

**I**) the additional node 2 (shown as diamond) is matched to a duplicated node 1a (shown as circle). This means the duplicated node 1b must be matched to an additional node (either 3, 4 or 5) as well. (

**II**) When 1b is matched, e.g., with node 3, the other additional nodes connected to 1a or 1b must be matched with additional nodes as well because 1a and 1b are matched already. (

**III**) It is possible that around nodes 1a,1b all additional nodes are matched with additional nodes. In this case, 1a and 1b are matched with each other.

**Figure 13.**Illustration of the algorithmic procedure for s-t paths: (

**I**) auxiliary graph ${G}_{\mathrm{A}}$. A modified mapping is used to create a minimum-weight path which connects the two nodes s to t. These two nodes are not duplicated. Black edges obtain the same weight values (partially shown as numbers here) as the corresponding edge in the original graph. Edges with zero weight are shown in grey (weight value 0 not shown), (

**II**) resulting MWPM, (

**III**) resulting path of minimum weight (bold edges) which is obtained from the MWPM on ${G}_{\mathrm{A}}$. For the example shown here, the path exhibits the weight ${\omega}_{p}=-1$. In addition, there are no additional loops in the graph but in principle they may be present.

**Figure 14.**Examples for configuration which consist of a path of minimum weight plus loops. Here, a square lattice of size $L\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}64$ is shown. Dashed (solid) lines at the boundary represent free (periodic) boundary conditions. The MWP is shown in black. The selected nodes s and t are drawn with black dots. Gray lines are used to show the loops. The snapshots are taken at different values $\rho $ for the disorder parameter; here (

**a**) $\rho =0.28<{\rho}_{\mathrm{c}};$ (

**b**) $\rho \approx {\rho}_{\mathrm{c}}=0.340(1)$; (

**c**) $\rho =0.7$, and (

**d**) full coverage $\rho =1$. The bottom scale indicates where the different configurations are located along the “disorder axis” in the range $\rho =0\dots 1$.

**Figure 15.**(

**I**) Example for a minimum-weight s-t path (bold lines); (

**II**) When shifting up all weight values, such that all weights are not negative, the shortest path changes.

**Figure 16.**Illustration of the construction of the auxiliary graph ${G}_{\mathrm{A}}$ for directed graphs G: (

**I**) Node in the original graph with two incoming and two outgoing edges. (

**II**) corresponding auxiliary graph with a sample perfect matching, representing a path using one ingoing and one outgoing edge.

**Figure 17.**Probability of a percolating loop for two-dimensional lattices with mixed Gaussian disorder according to Equation (4) in Section 4.1. The inset shows the probability ${P}_{L}^{s}$ that the system exhibits a system spanning loop as a function of the disorder parameter $\rho $, for different system sizes L. The main plot shows the same data with the $\rho $-axis rescaled to determine the critical point ${\rho}_{c}$ and the critical correlation length exponent $\nu $ via a data collapse according to Equation (5) of this section.

**Figure 18.**Distribution of the length of the non-percolating loops for two-dimensional NWP ($L=256$) with mixed Gaussian disorder according to Equation (4) for several values of the disorder parameter $\rho $. The lines show fits to the functions ${n}_{\ell}\sim {\ell}^{-\tau}exp(-{T}_{L}\ell )$. The inset shows the behaviour of ${T}_{L}$ as a function of the distance from the critical point.

**Figure 19.**Probability that a path connecting two “furthest” nodes in the $r=3$-regular graph have a negative weight, corresponding to the fact that they would percolate if only negative paths would be allowed. The inset shows the raw data, renormalized by the size-dependent critical point ${\rho}_{1}(N)$. The main plot displays the collapse of data yielding the value of critical exponent $\nu $.

**Figure 20.**The left-passage probability $p(\varphi )$ was obtained numerically for various points (257 different angles) on a semicircle ($R\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}128$). The simulations were performed right at the critical density for lattices with 1025 × 256 sites and averaged over 51,200 disorder realisations. The probability is shown together with the SLE prediction ${P}_{\kappa}(\varphi )$. The diffusion constant ${\kappa}^{\u2605}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3.343$ was chosen such that the agreement between ${P}_{\kappa}(\varphi )$ and $p(\varphi )$ is the largest. In the inset, the difference of the numerical left-passage probability to the prediction from SLE is shown, indicating that the agreement is indeed very good. To keep the inset clear, the difference is shown only for a selected number of typical data points.

**Figure 21.**

**Left**: determination of the correlations lengths ${\xi}_{\parallel}$ and ${\xi}_{\perp}$ which are parallel and perpendicular to the main (up to bottom) direction for directed NWP, respectively.

**Right**: correlation length ${\xi}_{\perp}$ for different ${L}_{\perp}$ at fixed ${L}_{\parallel}=256$ to determine $\xi ({L}_{\parallel}=256,{L}_{\perp}\to \infty )$. The inset shows the scaling of $\xi ({L}_{\parallel},{L}_{\perp}\to \infty )$ to find the ratio ${\nu}_{\parallel}/{\nu}_{\perp}$.

**Table 1.**Critical points ${\rho}_{c}$ and critical exponents $\nu $, $\beta $, $\gamma $, ${d}_{f}$, $\tau $, $\sigma $ for various dimensions d.

d | ${\mathit{\rho}}_{\mathit{c}}$ | $\mathit{\nu}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | ${\mathit{d}}_{\mathit{f}}$ | $\mathit{\tau}$ | $\mathit{\sigma}$ |
---|---|---|---|---|---|---|---|

2 | 0.340(1) | 1.49(7) | 1.07(6) | 0.77(7) | 1.266(2) | 2.59(3) | 0.53(3) |

3 | 0.1273(3) | 1.00(2) | 1.54(5) | −0.09(3) | 1.459(3) | 3.07(1) | 0.71(1) |

4 | 0.0640(2) | 0.80(3) | 1.91(11) | −0.66(5) | 1.60(1) | 3.55(2) | 0.78(2) |

5 | 0.0385(2) | 0.66(2) | 2.10(12) | −1.06(7) | 1.75(3) | 3.86(3) | 0.88(2) |

6 | 0.0265(2) | 0.50(1) | 1.92(6) | −0.99(3) | 2.00(1) | 4.00(2) | 0.97(4) |

7 | 0.0198(1) | 0.48(1) | – | – | 2.08(8) | 4.50(1) | – |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Hartmann, A.K.; Melchert, O.; Norrenbrock, C. From Spin Glasses to Negative-Weight Percolation. *Entropy* **2019**, *21*, 193.
https://doi.org/10.3390/e21020193

**AMA Style**

Hartmann AK, Melchert O, Norrenbrock C. From Spin Glasses to Negative-Weight Percolation. *Entropy*. 2019; 21(2):193.
https://doi.org/10.3390/e21020193

**Chicago/Turabian Style**

Hartmann, Alexander K., Oliver Melchert, and Christoph Norrenbrock. 2019. "From Spin Glasses to Negative-Weight Percolation" *Entropy* 21, no. 2: 193.
https://doi.org/10.3390/e21020193