Low Temperature Phase of the m-component Spin Glass

Abstract

We investigate the low temperature phase of a vector spin glass in the limit of large component numbers $m$ of the spins, where this model is expected to be replica symmetric. On a Bethe lattice, a random graph with fixed and finite connectivity, we employ a self-consistent effective-field approach to the cavity method to describe equilibrium states of the system in terms of the configuration of cavity fields, which are reduced mean fields onto each spin. We make use of an iterative algorithm to find such equilibrium states for finite system sizes $N$ and observe a phase transition, which is indicated by an order parameter describing the preferred direction each spin points into in terms of the configuration of cavity, resp. mean local fields. On single realizations of Bethe lattices the transition is sharp and accompanied by the Generalized-Bose-Einstein condensation, in which the cavity, resp. local fields of the spins condense into a subspace of the available $m$-dimensional space for large $m$. The dimension of this subspace, $m_{\rm eff}(T)$ depends on the temperature. Is either one or two right at the critical temperature and increases below. Furthermore, for fixed temperature it is expected to scale with the system size $N$ with an exponent $\mu_m$ which depends on the connectivity. Averaging over many realizations of disorder we find $\overline{m_{\rm eff}}(T_c)=2$ and $\overline{m_{\rm eff}}(T)\sim N^{\mu_m}$. Even more, the cavity fields are subject to rotations in the low temperature phase, i.e. there is no freezing in time as generally expected. In case of $m_{\rm eff}(T_c)=2$ the whole configuration rotates at the critical temperature about the same angle and the same axis of rotation corresponding to a single droplet. In addition, we find replica symmetry breaking below the large-$m$ limit, i.e. for $m_{\rm eff}(T) > m$. These results are supported by results of parallel tempering Monte Carlo simulations. Replica symmetry breaking is further investigated by an analytical method in the large connectivity limit, where we compare the free energy in terms of spin-spin-correlations of different replicas calculated with a replica symmetric and a one-step broken replica symmetry ansatz. In the large-$m$ limit, we find that both solutions coexist for finite connectivities and that replica symmetry holds for full connectivity up to corrections in second order around the saddle point for $m\to\infty$. In the last chapter we calculate the sample-to-sample fluctuations of the free energy of the SK model of the large-$m$ limit. We derive an exact connection between these fluctuations and bond chaos. Making use of a replica calculation for the {\it large deviations} and a scaling ansatz for the {\it small deviations}, we quantify bond chaos, i.e. calculate the probability of finding a given link overlap for two systems with possibly different sets of bonds. With this, we find the sample-to-sample fluctuations to scale with the number of spins, $N$, with an exponent $\mu$ in between $1/5$ and $3/10$.