Lecture 11: Magnetic Anisotropies Naoki KAWASHIMA ISSP, U. Tokyo July 1, 2019 Naoki KAWASHIMA  (ISSP) July 1, 2019 1 / 1 In this lecture, we see …. ● It is not only O(n) models that we can study by considering the multiple-component field. We can deal with anisotropies as well.
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● Real magnetic systems can never be truely isotropic because spins are coupled with orbital degrees of freedom that are subject to the influence of the lattice. ● In the case of the cubic lattice, for example, the localized spins feel the anisotropy field that has the same symmetry as the cubic lattice.
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● To understand why this term represents the effect of the cubic lattice, consider the case where ν → ∞. In this limit, the spin has to point to one of the corners of the unit cell (cube). ● Note that in this limit, the system becomes 3 decoupled Ising models.  We will find a fixed point corresponding to this limit.
Nacki KAWASHIMA (ISSP
Statistical Machanics I
July 1, 2019

● For the ε-expansion of the systems with the cubic symmetry, we consider [[· · ]] of each term in the Hamiltonian. ● t-operator: ● u-operator: ● v-operator:

Statistical Machanics
OPE ● φξφυ ≈ · · + 8φυ + 4( n + 2)φξ + ···
● φέφν ≈ · ·· + 8φν + 12φε + ···
● φυφν ≈ · · + 24φυ + 48φν + 96φt + ···

Statistical Machanics I
July 1, 2019 6 / 1 RG flow equation ● Keeping in mind that u = O(ε) and t = O(ε²), as before, the part of the RG flow equation necessary for the lowest order discussion is
Note that we have omitted the terms, such as tu in B and u 2 in A, that would not contribute to yt, yu, yv at the non-Gaussian FPs. ● We have four fixed points: (t,u,v) = (0,0,0) ("decoupled Ising FP" ) ( toi, 0, voi) (t) . U).
Statistical Machanics I
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Linearization
● In all cases, we have the same form of the linearized RG flow eqs. in terms of Δu ≡ (t − t*, u − u*, v − v*)T:
● The lower-right 2 × 2 sub-matrix is important:

Naoki KAWASHIMA  (ISSP)
Statistical Machanics I
July 1, 2019
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● The eigenvalues and eigenvectors are ● Therefore, we have n c ≈ 4 and the WFFP is stable if n < nc. Case 1: n < nc Case 2: n > nc
Statistical Machanics I
July 1, 2019

● Remembering the RG flow equation for ν, we find a FP with u* = 0:
● The eigenvalues and eigenvectors are
Nacki KAWASHIMA (ISSP
Statistical Machanics I
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“C” ... Cubic fixed point
Case 1: n < nc Case 2: n > n受
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● Putting together, we can draw the RG flow diagram including the 4 fixed points.
● Depending on whether n < n c or n > n c we can draw two types of the diagram. ● So, after all the cubic anisotropy is irrelevant for real magnetic systems? Case 1: n < nc Case 2: n > nc
Statistical Machanics I
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● The value for n ς seems to be close to 3 in 3D. So, there has been a long-standing controversy about the nature of the ferromagnetic transition under the cubic anisotropy. ● According to [Varnashev: PRB 61 14660 (2000)] nc(d = 3) < 3, or more specifically  nc (d = 3) = 2.89(2). ● For general discussion see [Calabrese et al: arXiv:cond-mat/0509415].
Statistical Machanics I
● By representing the cubic anisotropy by the term ν ∑ α (φ ῖ ) 4 , we have constructed a field theory that may explain the effect of the lattice anisotropy on the spin systems that is otherwise symmetric. ● The ε-expansion of the φ4 model with the ν term produces a new fixed point. (Cubic fixed point) ● The cubic fixed point is stable for n > nc whereas it is unstable for n < nς, where nς = 4 + O(ϵ). ● According to a more sophisticated numerical estimate, nc in 3D is slightly below 3, which suggests that we cannot simply neglect the cubic anisotropy in 3D. ● However, the critical region may be narrow in real systems due to smallness of the cubic anisotropy field and the proximity of nc to 3.
● Consider the critical point of the Heisenberg model. Discuss the effect of the uniaxial symmetry breaking-field that is represented by adding the term
to the isotropic Hamiltonian, i.e., the regular Heisenberg model. (Consider the scaling dimension of the scaling operator that corresponds to the above operator, and obtain its scaling dimension at the Wilson-Fisher fixed point for n = 3, to the lowest order in ε .) 15 / 1