Frustrated magnets in the limit of infinite dimensions: Dynamics and disorder-free glass transition

We study the statistical mechanics and the equilibrium dynamics of a system of classical Heisenberg spins with frustrated interactions on a $d$-dimensional simple hypercubic lattice, in the limit of infinite dimensionality $d\to \infty$. In the analysis we consider a class of models in which the matrix of exchange constants is a linear combination of powers of the adjacency matrix. This choice leads to a special property: the Fourier transform of the exchange coupling $J\left(\mathbf{k}\right)$ presents a $(d-1)$-dimensional surface of degenerate maxima in momentum space. Using the cavity method, we find that the statistical mechanics of the system presents for $d\to \infty$ a paramagnetic solution which remains locally stable at all temperatures down to $T=0$. To investigate whether the system undergoes a glass transition we study its dynamical properties assuming a purely dissipative Langevin equation, and mapping the system to an effective single-spin problem subject to a colored Gaussian noise. The conditions under which a glass transition occurs are discussed including the possibility of a local anisotropy and a simple type of anisotropic exchange. The general results are applied explicitly to a simple model, equivalent to the isotropic Heisenberg antiferromagnet on the $d$-dimensional face-centered-cubic lattice with first- and second-nearest-neighbor interactions tuned to the point $J_1=2J_2$. In this model, we find a dynamical glass transition at a temperature $T_{\mathrm{g}}$ separating a high-temperature liquid phase and a low-temperature vitrified phase. At the dynamical transition, the Edwards-Anderson order parameter presents a jump demonstrating a first-order phase transition.

Figure 1

Partitioning of the cubic lattice into two interpenetrating fcc sublattices ($A$ and $B$) for $d=3$. The sublattices $A$ and $B$ can be identified as the sets of lattice points $(x_1,\cdots ,x_d)$ for which ${\sum }_{a=1}^d{x}_a$ is, respectively, even and odd. The separation of even and odd sites can be applied in arbitrary dimension $d$ to separate the hypercubic lattice into fcc sublattices [53].

Figure 2

Temperature dependence of the variance of the cavity distribution $L$ (dashed line) and of the full equilibrium distribution $\lambda$ (solid line), in the isotropic model with $V\left(\mathbf{S}\right)=0, f^{\alpha \beta }\left(x\right)=J(x^2-1){\delta }^{\alpha \beta }$. The top panel shows the low-temperature region, the bottom panel a wider range of temperatures. At low $T, L\left(T\right)\approx k_{\mathrm{B}}T{f}_{\max }=k_{\mathrm{B}}T\left|J\right|$. (The dotted line is a guide to the eye highlighting the asymptotic behavior at $T\to 0$.) The variance $\lambda$, instead, approaches a finite limit $f_{\max }^2/\left(3J^2\right)$. The temperature dependence $\lambda \left(T\right)$ is nonmonotonic: $\lambda \left(T\right)$ has a minimum at $T\simeq 0.167\left|J\right|$. At higher temperatures the curves $L\left(T\right)$ and $\lambda \left(T\right)$ approach each other, and eventually saturate to $L\left(T\right)\simeq \lambda \left(T\right)\simeq 2J^2/3$ in the limit $T\to \infty$.

Figure 3

Equilibrium distribution of the field ${\mathbf{b}}_i$ for 10 different temperatures, equally spaced between ${10}^{-3}\left|J\right|/k_{\mathrm{B}}$ and $0.25\left|J\right|/k_{\mathrm{B}}$ (solid lines). The 10 values of the temperature are, with three-digit precision, $k_{\mathrm{B}}T/\left|J\right|$ = 0.001, 0.029, 0.056, 0.084, 0.112, 0.139, 0.167, 0.195, 0.222, 0.25. The dashed line shows the equilibrium distribution of the field ${\mathbf{b}}_i$ at the glass temperature $T_{\mathrm{g}}\simeq 0.0103\left|J\right|/k_{\mathrm{B}}$. For the higher temperatures in the plot, the distribution differs from a Gaussian, but has a bell-like shape with a maximum at $b_i=0$. At lower temperatures, the maximum shifts away from $b_i=0$. For $T\to 0$, the distribution becomes concentrated on a narrow shell near the spherical surface $|{\mathbf{b}}_i|=|J|=f_{\max }$.

Figure 4

Numerical analysis of Eqs. (75). The curves show the function $I(\beta /{\sigma }_1)/\beta -I_1\left({\beta }^2{L}_2\right)/3$ calculated substituting $\beta /{\sigma }_1=-1/\left(\beta c_1\right)-\beta L_1=-3/\left[\beta I_1\left({\beta }^2{L}_2\right)\right]+\beta L_2-\beta L$. The $x$ axis is parametrized by the order parameter $q_{\mathrm{EA}}\left({\beta }^2{L}_2\right)=[1-I_1\left({\beta }^2{L}_2\right)]/3$. The dotted, solid, and dashed curves represent the curves calculated at temperatures respectively equal to $T=0.012\left|J\right|/k_{\mathrm{B}}, 0.0103\left|J\right|/k_{\mathrm{B}}$, and $0.009\left|J\right|/k_{\mathrm{B}}$. The solutions of the self-consistency problem are defined by the points at which the curves cross 0. For high temperature the curve intersects 0 only at the trivial point $q_{\mathrm{EA}}=0$ (dotted line). A nontrivial solution appears at $T_{\mathrm{g}}\simeq 0.0103\left|J\right|/k_{\mathrm{B}}$. Below $T_{\mathrm{g}}$, the equations have two solutions $q_{\mathrm{EA}\pm }$. The physical solution is $q_{\mathrm{EA}+}$, the one with the largest order parameter.

Figure 5

Equilibrium average of the Edwards-Anderson order parameter $q_{\mathrm{EA}}\left(T\right)$ obtained from a numerical solution of Eqs. (75). The order parameter represented in the figure describes the large-time limit $3q_{\mathrm{EA}}\left(T\right)={lim}_{|t-t^{\prime }|\to \infty }\langle S^{\alpha }\left(t\right)S^{\alpha }\left(t^{\prime }\right)\rangle$, with an averaging $\langle \cdots \rangle$ weighted with the equilibrium Gibbs distribution at temperature $T$. This equilibrium $q_{\mathrm{EA}}$ may differ in general from the average order parameter which the system presents after a cooling from the paramagnetic phase $T>T_{\mathrm{g}}$ because the spins remain trapped in a single metastable state at $T_{\mathrm{g}}$ and fall out of equilibrium at lower temperatures.