The nature of the ordered state of spin glasses

The nature of the ordered phase of spin glasses has been controversial for four decades. There are 4 distinct theories of the ordered phase: the broken replica symmetry picture of Parisi, the droplet scaling picture, the chaotic pairs theory and the TNT picture. These theories differ in their predictions of the nature of droplets of reversed spins. For the Parisi RSB theory such droplets are space filling with surface fractal dimension $d_s=d$ and cost a (free) energy of $O(1)$. In the droplet scaling picture the droplets have a fractal dimension $d_s < d$ and have a free energy of $L^{\theta^{\prime}}$. In the chaotic pairs theory the droplets have a free energy cost as in droplet scaling but they are space filling with $d_s = d$. The TNT picture has droplets whose free energies are of $O(1)$ as for the RSB theory but with a fractal dimension $d_s < d$.

In an applied field there is on the RSB picture and the chaotic pairs theory a phase transition on cooling at the de Almeida-Thouless (AT) line to a state with replica symmetry breaking. For the droplet and TNT pictures there is no transition in the presence of a field, just as happens for a ferromagnet in a field. There are old arguments which suggest that the AT line only exists above 6 dimensions. I shall review these arguments and then describe numerical simulations whose purpose is to find if the AT line goes away as $d \to 6$. Simulations in such high dimensions are very challenging, so instead I will describe simulations on a proxy model,  the one-dimensional power-law diluted  spin glass model, in which the probability that two spins separated by a distance $r$ interact with each other, decays as $1/r^{2\sigma}$. Tuning the exponent $\sigma$ is equivalent to changing the space dimension of a short-range model Edwards-Anderson model. We have studied the form of the AT line in the Heisenberg spin glass and have found evidence that the AT line will indeed disappear below six dimensions, as $h_{AT}^2 \sim (d-6)$.

That leaves just the droplet and TNT pictures as candidate theories for three dimensions. We have some tentative evidence that the features associated with the TNT picture -- droplets of the size of the system of free energy cost of $O(1)$ -- are just consequences of studying systems which are too small, and that TNT features  go away as the system size is increased.  This leaves the droplet scaling theory as the correct picture for three dimensional spin glasses.

[Work done in collaboration with Bharadwaj Vedula and Auditya Sharma, Bhopal, India]