Wolfram Alpha:

Goldstone Modes

To see how the disorder a↵ects the fluctuation-stabilised spiral phase, we need to examine its e↵ects on the Goldstone modes of the system. The approach thus far has treated the Goldstone modes in a slightly naive way, and has assumed the Hamiltonian to be invariant under rotation of the spiral ordering wavevector.

The resulting Goldstone modes associated with such rotations are artefacts of this theoretical approach. In any real system, there will be anisotropies generated by the crystal field which, among other e↵ects, will act to pin the spiral ordering wavevector along a particular lattice direction. Here, we add a cubic anisotropy term of the following form:

Fan =

Z

d3r X

↵=x,y,z

(@↵m↵)2. (2.76)

dispersion at low filling fractions. For > 0 this is minimised by having the spiral wavevector point along one of the equivalent crystal axes. This pinning along a crystal axis does not change the form of the saddle-point equation used earlier to minimiseq. Here, as before, we chooseq=qzˆwithout loss of generality. Now that we have restricted the direction of the spiral ordering wavevector, we have eliminated the Goldstone modes associated with continuous rotations of the direction of q. The only remaining Goldstone modes are rotations in the phase of the ordering wavevector. The spiral order parameter becomes:

m(r) = m[ˆxcos (qz+ (r)) + ˆysin (qz+ (r))]. (2.77) The Goldstone modes of the spiral phase correspond to continuous deformations

(r) of the phase of the spiral order parameter.

By expanding the action around the saddle point valueq2 _{= f}

2(m2)/f4(m2)

in terms of this phase slip (r), we find that these deformations can be captured by a classical, anisotropic 3d-XY action of the following form:

S _⇡ Z d3r " 2f4(m2)q2(@z )2+ 1 2 m 2 X i=x,y (@i )2 # , (2.78)

which can be derived by re-inserting Eq. 2.77 into the free energy expression in Eq. 2.31, after re-expressing the q2 _{terms in the free energy as gradient terms}

and allowing for a spatially varying phase (r).

The charge disorder that we consider here generates a random mass disorder for the ferromagnet, i.e. it appears in the coefficient of them2 _{term in the Landau}

expansion. This disorder leads to some quantitative changes but otherwise has no e↵ect on the long-range magnetic correlations in the ferromagnetic phase. However, from the saddle point equation for q, we see that what is a relatively weak random mass disorder in the ferromagnetic phase has a much stronger e↵ect in the spiral, inducing a spatial variation in the spiral ordering wavevector q.

In conjunction with the cubic anisotropy, this disorder in the pitch of the spiral leads to a much stronger random anisotropy disorder in the Goldstone modes (r). We can see this by plugging Eq. 2.77 into the anisotropy term Eq. 2.76, allowing for variations in the pitch of the spiralq !q+ q(r) and expanding in powers of q. The net result of this procedure is that the disorder generates a random anisotropy term for the phase of the spiral:

Sdis = 1 2 m

2

Z

d3rg(r) cos [2 +↵(r)], (2.79) where ↵(r) _⇠ q(r)z is the random phase and g(r) = (1/4)[(@y↵)2 (@x↵)2].

This random anisotropy is only induced in the fluctuation-stabilised phase, not in the homogeneous ferromagnet. This demonstrates that the disorder has a distinctly di↵erent - and much stronger - e↵ect on the spiral phase than on the

ferromagnet. It has previously been shown that random anisotropy disorder of this type is sufficient to destroy long range order in dimensions d < 4 [108–110]. The disordered spiral phase is therefore not a long-range ordered phase.

It is possible that the system may display quasi-long-range-order, though this is unlikely [111]. Weak random anisotropy disorder [112] in O(N) magnets with N < Nc = 9.44 has been shown to lead to algebraic quasi-long-range-order for

dlc < d < 4 with a lower critical dimension dlc ⇡ 4 0.00158(N Nc)2. In the

case of the XY system considered here, dlc(N = 2) ⇡3.91. Similar lower critical

dimensions have been found using non-perturbative functional renormalisation group for the case of random field disorder [113], adding up to the conclusion that it is very likely the disordered spiral displays short-ranged magnetic correlations.

Correlation Length

To obtain an expression for the correlation length, we use the result of Ref. [114] which shows that the correlation length of a 3d-XY model with random anisotropy disorder may be written as:

⇠= n 2_⇢2 s 8⇡D2_K2 n , (2.80)

where ⇢s is the spin sti↵ness, the random anisotropy disorder takes the form

Dcos(n +↵(r)) and Kn = hcos(n )i0 is computed in the absence of disorder.

For the random anisotropy we consider here, n = 2. We may extend this result to our anisotropic 3d-XY model by rescaling the z co-ordinate, allowing us to write down an expression for the correlation length in the plane perpendicular to

q and along the direction of the spiral:

⇠x,y = 1 2⇡ 2 g ✓ m2 4f4(m2)q2 ◆2 exp T˜ ⇡p4 f4(m2)m2q2 ! , (2.81) ⇠z = 1 2⇡ 2 g ✓ m2 4f4(m2)q2 ◆5/2 exp T˜ ⇡p4 f4(m2)m2q2 ! . (2.82) Note that in the limit of zero disorder, 2

g ! 0, the correlation length diverges

and long range order is recovered. This correlation length is highly anisotropic, with a ratio given by:

⇠z ⇠xy = r 4f4(m2)q2 m2 . (2.83)

Figure 2.8 shows the pronounced anisotropy of the correlation length. The anisotropy is not entirely a surprise, since the ‘background’ ordering of the phase is a spiral ferromagnet, but the strong dependence on both temperature and interaction strength is unusual. The same is true for the ordering wavevector.

This potentially o↵ers a new avenue for the detection of helical ordering in addition to, or instead of, neutron scattering. Establishing the presence of such a correlation length (in the disordered system only) and/or ordering wavevector that scales so strongly with temperature and some relevant experimental tuning parameter would be a strong indication for the presence of this type of phase.