# Spin Glasses in a Field Show a Phase Transition Varying the Distance among Real Replicas (And How to Exploit It to Find the Critical Line in a Field)

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## Abstract

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## 1. Introduction

## 2. Phase Transition Varying the Overlap between Two Real Replicas in a Solvable Mean Field Model

#### 2.1. The Truncated Model

#### 2.2. The Model with Constrained Replicas

- If ${q}^{\prime}\left(x\right)=0$, then ${p}^{\prime}\left(x\right)=0$.
- If ${q}^{\prime}\left(x\right)\ne 0$ and ${p}^{\prime}\left(x\right)=0$, then $q\left(x\right)=\frac{x}{3y}$.
- If $p\left(z\right)=q\left(z\right)\phantom{\rule{0.277778em}{0ex}}\forall z\le x$, then either ${q}^{\prime}\left(x\right)=0$ or $q\left(x\right)=\frac{2x}{3y}$.

#### 2.3. Replica Symmetry (RS) Solutions

#### 2.4. Replica Symmetry Breaking (RSB) Solutions in the Paramagnetic Phase

- a solution with ${p}_{0}\gtrsim {p}_{1}\simeq p$ and ${q}_{0}\lesssim {q}_{1}\simeq q$, i.e., with the $p\left(x\right)$ and $q\left(x\right)$, respectively, very close to the RS corresponding overlaps p and q,
- a solution with ${p}_{1}\simeq p<{p}_{0}={q}_{0}<{q}_{1}\simeq q$, i.e., where ${p}_{1}$ and ${q}_{1}$ are close to the RS overlaps and at a small x, a mean overlap is roughly found ${p}_{0}={q}_{0}\simeq \frac{p+q}{2}$.

## 3. Numerical Results in a Finite-Dimensional Spin Glass Model Varying the Overlap between Two Real Replicas

#### 3.1. Model and Numerical Simulations

#### 3.2. A New Tool of Analysis Conditioning on the Overlap

#### 3.3. Numerical Results

- from the peak location in $P\left(q\right)$, and
- from the crossing points of the cumulative functions $x\left(q\right)\equiv {\int}_{-1}^{q}d{q}^{\prime}\phantom{\rule{0.166667em}{0ex}}P\left({q}^{\prime}\right)$.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Parameters of the RS solutions versus ${p}_{d}$ in the case of $\tau =h=0.1$ and $y=2/3$. The merging of the three curves takes place at ${p}_{d}={p}_{d}^{*}(\tau ,h,y)=0.117033$, while the crossing between the two curves takes place at ${q}_{\mathrm{EA}}(\tau ,h,y)=0.141942$.

**Figure 2.**Values of ${q}_{\mathrm{EA}}(\tau ,h,y)$ and ${p}_{d}^{*}(\tau ,h,y)$ plotted in the $(h,T=1-\tau )$ plane for $y=2/3$. The red bold curve is the dAT line, separating the paramagnetic and the spin glass phases. ${q}_{\mathrm{EA}}$ and ${p}_{d}^{*}$ merge on the dAT line, while their values in the spin glass phase have no physical meaning. Below the blue surface in the paramagnetic phase, the $p\left(x\right)=q\left(x\right)$ symmetry is broken.

**Figure 3.**Free energies of the $p=q$ and $p\ne q$ RS solutions for $\tau =h=0.1$ and $y=2/3$. Below ${p}_{d}^{*}=0.117033$, the free energy of the $p\ne q$ solution is higher, and such a solution dominates over the symmetric one (

**left panel**). The free energy difference goes as ${({p}_{d}^{*}-{p}_{d})}^{3}$, as can be seen in the (

**right panel**), where the black dot marks the value of ${p}_{d}^{*}$.

**Figure 4.**Difference between the one-step RSB (1RSB) free energies and ${F}_{\mathrm{RS}}$ for the two solutions with $\tau =h=0.1$, $y=2/3$, and ${p}_{d}=0.05$. We notice that the difference is very small, but clearly non-zero. Moreover, the maximum is achieved for a rather small value of m, thus limiting the difference with respect to the RS solution to very small values of x (we remind the reader that, in both 1RSB solutions, ${p}_{1}\simeq p$ and ${q}_{1}\simeq q$).

**Figure 5.**Difference between the dominating 1RSB free energy and ${F}_{\mathrm{RS}}$ as a function of m (

**left**) and ${p}_{d}$ (

**right**). The left panel shows that the location of the maximum of ${F}_{\mathrm{RSB}}$ slightly decreases when ${p}_{d}$ grows, but the main effect is that, for any m value, ${F}_{\mathrm{RSB}}$ tends to move toward ${F}_{\mathrm{RS}}$ when ${p}_{d}$ grows. The right panel shows that, for different m values, the free energy difference becomes zero very close to ${p}_{d}^{*}$, marked with a black dot. Note that data in the region close to ${p}_{d}^{*}$ may have some uncertainty due to the extremely small free energy differences, which are of the order $O\left({10}^{-12}\right)$.

**Figure 6.**$\chi \left(q\right)/{L}^{2-\eta}$ versus q for $\rho =1.4$ (non-mean field region) and six different lattice sizes. Data in the upper panels have been measured with $T=1.7$ and $h=0.2$ and belong to the paramagnetic phase [16], thus showing that a transition to a spin glass phase can be induced merely by decreasing the overlap between the replicas. In the bottom panels, near or inside the thermodynamic spin glass phase, $T=1.2$ and $h=0.2$. The crossing point of the curves for different lattice sizes is always very neat, as can be appreciated from the panels on the right that zoom in on the crossing region.

**Figure 7.**The cumulative probability distribution $x\left(q\right)$ versus q for $\rho =1.4$ (non-mean field region), $h=0.2$, and two values of the temperature: $T=1.7$ (

**left panel**) and $T=1.2$ (

**right panel**). The estimate for ${q}_{\mathrm{EA}}$ comes from the crossing of these curves.

**Figure 8.**Behavior of ${q}_{c}\left(T\right)$ and ${q}_{\mathrm{EA}}\left(T\right)$ for $\rho =1.4$ (

**top-left panel**with $h=0.3$,

**middle-left panel**with $h=0.2$ and

**bottom-left**with $h=0.1$) in the non.mean field regime and $\rho =1.2$ (

**top-right panel**with $h=0.3$ and

**bottom-right panel**with $h=0.2$) in the mean field regime. The crossing (or merging) of the curves identifies the thermodynamic phase transition to the spin glass phase (dAT line) because ${q}_{c}<{q}_{\mathrm{EA}}$ holds in the paramagnetic phase. Data shown are for the largest sizes ($L={2}^{12}$ and $L={2}^{13}$).

**Table 1.**Values of the critical temperature obtained from the crossing points of the curves ${q}_{\mathrm{EA}}\left(T\right)$ and ${q}_{c}\left(T\right)$, which have been computed using data from lattices L and $2L$. The left table is for $\rho =1.4$ (non-mean field regime), and the right table is for $\rho =1.2$ (mean field regime). In the row labeled FSSA, we report the critical temperatures obtained in [16] using finite size scaling analysis.

$\mathit{\rho}=1.4$ | $\mathit{\rho}=1.2$ | |||||
---|---|---|---|---|---|---|

${log}_{\mathbf{2}}\mathit{L}$ | $\mathit{h}=\mathbf{0.1}$ | $\mathit{h}=\mathbf{0.2}$ | $\mathit{h}=\mathbf{0.3}$ | ${log}_{\mathbf{2}}\mathit{L}$ | $\mathit{h}=\mathbf{0.2}$ | $\mathit{h}=\mathbf{0.3}$ |

${\mathit{T}}_{\mathit{c}}$ | ${\mathit{T}}_{\mathit{c}}$ | ${\mathit{T}}_{\mathit{c}}$ | ${\mathit{T}}_{\mathit{c}}$ | ${\mathit{T}}_{\mathit{c}}$ | ||

8 | 1.88(1) | 1.56(6) | 1.31(4) | 8 | 1.47(10) | |

9 | 1.89(3) | 1.44(6) | 1.39(3) | 9 | 1.36(5) | 1.38(5) |

10 | 1.85(1) | 1.47(2) | 1.40(1) | 10 | 1.4(1) | 1.43(4) |

11 | 1.40(3) | 1.53(1) | 1.39(3) | 11 | 1.48(5) | 1.47(3) |

12 | 1.57(9) | 1.51(1) | 1.37(1) | 12 | 1.51(5) | 1.53(2) |

FSSA | 1.67(7) | 1.2(2) | FSSA | 1.4(2) | 1.5(4) |

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**MDPI and ACS Style**

Dilucca, M.; Leuzzi, L.; Parisi, G.; Ricci-Tersenghi, F.; Ruiz-Lorenzo, J.J.
Spin Glasses in a Field Show a Phase Transition Varying the Distance among Real Replicas (And How to Exploit It to Find the Critical Line in a Field). *Entropy* **2020**, *22*, 250.
https://doi.org/10.3390/e22020250

**AMA Style**

Dilucca M, Leuzzi L, Parisi G, Ricci-Tersenghi F, Ruiz-Lorenzo JJ.
Spin Glasses in a Field Show a Phase Transition Varying the Distance among Real Replicas (And How to Exploit It to Find the Critical Line in a Field). *Entropy*. 2020; 22(2):250.
https://doi.org/10.3390/e22020250

**Chicago/Turabian Style**

Dilucca, Maddalena, Luca Leuzzi, Giorgio Parisi, Federico Ricci-Tersenghi, and Juan J. Ruiz-Lorenzo.
2020. "Spin Glasses in a Field Show a Phase Transition Varying the Distance among Real Replicas (And How to Exploit It to Find the Critical Line in a Field)" *Entropy* 22, no. 2: 250.
https://doi.org/10.3390/e22020250