Evidence of de Almeida-Thouless line below six dimensions

TL;DR

This study investigates the existence and nature of the de Almeida–Thouless line in five dimensions by performing large-scale Monte Carlo simulations of the Edwards–Anderson Ising spin glass with and without an external field, using finite-size scaling to extract critical properties. It combines analyses of replicon and anomalous susceptibilities, the second-moment correlation length, and RS effective-Hamiltonian observables (notably $\lambda_r$) to characterize the phase transitions and scaling corrections; crucially, it avoids zero-momentum pathologies by employing nonzero-momentum observables and zero-mode extrapolations. The results provide clear evidence for a dAT line in $d=5$, with a zero-field lower critical dimension estimated at $d_L^{h=0} \approx 2.43$, and yield consistent estimates for critical exponents (e.g., $\nu \approx 0.73$, $\eta \approx -0.21$ at $h=0$) and a continuous transition in a field ($\lambda_r(T_c^+) \approx 0.53$). These findings help bridge the gap between results in lower ($d=3$–$4$) and higher ($d=6$) dimensions and illuminate finite-field scaling behavior and the role of the zero-momentum mode in spin-glass criticality.

Abstract

We study the critical behavior of the Ising spin glass in five spatial dimensions through large-scale Monte Carlo simulations and finite-size scaling analysis. Numerical evidence for a phase transition is found both with and without an externally applied magnetic field. The critical exponents are computed in both cases. We compute with a 10% accuracy the lower critical dimension at zero magnetic field, finding a result consistent with estimates obtained with entirely different methods, by combining our estimates of critical exponents in five dimensions with previous results for other spatial dimensions. When the results in a magnetic field are compared with previous results in six spatial dimensions, qualitative differences emerge in the scaling behavior of the correlation functions at zero external momentum. This anomalous scaling does not extend to other wavevectors. We do not find indications of a quasi first-order phase transition in a magnetic field.

Figures (9)

• Figure 1: Second moment correlation length $\xi_2$, see Eq. \ref{['xi2']}, measured in units of the lattice size $L$ (top) and the dimensionless quotient $R_{12}$, see Eq. \ref{['eq:R12']}, (bottom), as a function of temperature $T$ for lattice sizes ranging from $L=6$ to $L=12$ and zero external magnetic field $h=0$.
• Figure 2: Quotient of $\xi/L$ at the crossing of $R_{12}$ and vice versa. The lines represent the joint fit to Eq. \ref{['eq:FSS_omega']} using both sets of data, which gives $\omega=3.7(4)$.
• Figure 3: Critical exponents $\eta$ and $\nu$ as a function of the dimension for $h=0$. The points correspond to the value of the critical exponents that appear in Table \ref{['tab:exponentes_vs_dim']}. The solid lines correspond to the fits to the functions $f^\nu_{\mathrm{fit}}$ and $f^\eta_{\mathrm{fit}}$ defined in the text. The horizontal dashed lines at zero are used for reference to the eye. Statistical errors are present but, in most cases, smaller than the symbol size.
• Figure 4: Replicon susceptibility $\chi_R$ (on the top), anomalous susceptibility (on the middle) and the quotient between the anomalous and replicon susceptibilities (on the bottom) as a function of temperature for different lattice sizes at $h=0.075$. The two vertical lines are our estimates for the critical temperature at $h=0$ (right grey line) and $h=0.075$ (left black line). See how $\chi_R$ grows rapidly with $L$ at low temperature, while $\chi_A/\chi_R$ goes to zero, ensuring that we are working far from the $h=0$ point of the de Almedia-Thouless line.
• Figure 5: Second moment correlation length $\xi_2$, see Eq. \ref{['xi2']}, measured in units of the lattice size $L$ (on the top) and the dimensionless quotient $R_{12}$, Eq. \ref{['eq:R12']} (on the bottom), as a function of temperature $T$ for lattice sizes ranging from $L=6$ to $L=10$ and non-zero external magnetic field $h=0.075$.