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# Glassy Freezing and Long-Range Order in the 3 D Random Field XY Model 

Ronald Fisch ∗ 382 Willowbrook Dr. North Brunswick, NJ 08902 

(Dated: November 3, 2018) 

Monte Carlo studies of the 3D random field XY model on simple cubic lattices of size 64 $^{3}$, using two different isotropic random-field probability distributions of moderate strength, show a glassy freezing behavior above T c , and long-range order below T c , consistent with our earlier results for weaker and stronger random fields. This model should describe random pinning in vortex lattices in type-II superconducting alloys, charge-density wave materials, and decagonal quasicrystals. 

The random-field XY model (RFXYM) was first stud- ied forty years ago,[1] as a toy model for random pinning effects in the vortex lattice of a type-II superconductor in an external magnetic field. For fixed-length classical spins the Hamiltonian of the RFXYM is 

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Each φ$_{i}$ is a dynamical variable which takes on values be- tween 0 and 2 π . The 〈 ij 〉 indicates here a sum over near- est neighbors on a simple cubic lattice of size L × L × L . We choose each θ$_{i}$ to be an independent identically dis- tributed quenched random variable, with the flat proba- bility distribution 

for θ$_{i}$ between 0 and 2 π . The probability distribu- tion P ( h$_{i}$ ) is independent of θ$_{i}$ . This Hamiltonian is closely related to models of incommensurate charge den- sity waves,[2, 3] which include decagonal and other axial quasicrystals[4] as special cases. In all of these cases, the physical origin of the “random field” is the fact that any real sample contains a finite density of point defects, i.e. atoms sitting in places where they would not be in the ideal structure. 

We will use the term “sample” to denote a Hamiltonian with a particular set of the h$_{i}$ and θ$_{i}$ variables. A thermal average of some quantity over a single sample will be denoted here by angle brackets, 〈 〉 , and an average over all samples will be denoted by square brackets, [ ]. 

Larkin[1] replaced the spin-exchange term of the Hamiltonian with a harmonic potential, so that each φ$_{i}$ is no longer restricted to lie in a compact interval. This approximation, which neglects mode coupling, is strictly valid only in the limit n → ∞ , where n is the number of components for each spin. Larkin then argued that, for any probability distribution P ( h$_{i}$ ) which allows non- zero values for h$_{i}$ , this model has no long-range ordered phase on a lattice whose spatial dimension d is less than or equal to four. 

A more intuitive derivation of this result was given by Imry and Ma,[5] who assumed that the increase in the energy of an L d lattice when the order parameter is 

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twisted at a boundary scales as L d − $^{2}$. This was later justified[7, 8] by showing that within a perturbative ϵ ex- pansion one finds the phenomenon of “dimensional reduc- tion”. Term by term for T > T$_{c}$ , the critical exponents of any d -dimensional O ( n ) random-field model appear to be identical to those of an ordinary O ( n ) model of dimension d − 2. 

However, reservations about the correctness of apply- ing the n = ∞ results for finite n , i.e. the substitution of the harmonic potential for the exchange term, were soon expressed by Grinstein.[6] For the Ising ( n = 1) case, dimensional reduction was shown rigorously to be incorrect.[9, 10] 

One problem with dimensional reduction for n > 1 is that the perturbation theory can only be calculated for T > T$_{c}$ , while the twist energy of the order parameter is only defined for T < T$_{c}$ . Another, more interesting problem for n > 1 is associated with the question of properly defining the order parameter. As discussed, for example, by Griffiths,[11] there is more than one way to do this. For the random-field model, there are several different types of two-point correlation functions.[6] 

When n = 2, one may use the vector order parameter ⃗ M , defined by 

where N = L d is the number of spins. Alternatively, one can choose the scalar order parameter M $^{2}$, defined by 

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It is very awkward to calculate ⃗ M , since its value will vary wildly from sample to sample. Thus we expect that [ ⃗ M ] will be zero, even though 〈 ⃗ M 〉 for each sample may become large at low temperature. On the other hand, we anticipate that for weak disorder the probability distri- bution for M 2 will become narrow as we take L → ∞ . Thus it is the quantity 〈 M $^{2}$〉 , and not the quantity 〈 ⃗ M 〉 , which is the natural carrier of long-range order in the presence of the random field. 

It is also necessary to consider the possibility that the thermal average 〈 M $^{2}$〉 for a typical finite sample may be- come large as T is lowered, even though 〈 ⃗ M 〉 remains 

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