-
15
-
-
4243981460
-
-
R. A. Pelcovits [, ] has performed a first-order 4+ epsilon expansion for the random second-rank anisotropy model discussed in Ref. 19 and the results have been generalized to long-range interactions by M. Chang and E. Abrahams , Phys. Rev. B, 27, 5570
-
(1979)
Phys. Rev. B
, vol.19
, pp. 465
-
-
-
20
-
-
84926604648
-
-
The dangerous irrelevancy of temperature in 6- epsilon dimensions causes, for example, the connected correlation functions to fall off more rapidly at the critical point than the full disconnected correlation functions.
-
-
-
-
23
-
-
84926583921
-
-
For the X-Y case it is more convenient to work with m-fold cos(m theta) anisotropies than the muth-rank tensor anisotropies introduced for the general case. This is because the symmetry group for the X-Y case is Abelian, and hence a single Γm can generate under renormalization only those Γk for which k is a multiple of m. Thus there are an infinite class of random cos(m theta) anisotropy (m is often denoted p;
-
-
-
-
24
-
-
84926548255
-
-
see, e.g., Ref. 21) models with different symmetry. This is not true for general n;
-
-
-
-
25
-
-
84926587878
-
-
the only symmetry restriction on the generation of other Δμ aposs by one of them is parity;
-
-
-
-
26
-
-
84926567659
-
-
i.e., even muth-rank anisotropy cannot generate odd muth-rank anisotropy. For the X-Y case Δmu can be written as a combination of all the Γm with μ - m even and non-negative as in Eq. (2.4). Note that we have called the general anisotropies muth rank to avoid confusion with m-fold anisotropies for the X-Y case which are just the Γm. Twofold anisotropy for X-Y spins is (up to an important constant) the same as second-rank anisotropy and we will use the terms interchangeably for this case.
-
-
-
-
27
-
-
0000340515
-
-
J. L. Cardy and S. Ostlund [, ] have studied the X-Y case with random fields and anisotropies in two dimensions.
-
(1982)
Phys. Rev. B
, vol.25
, pp. 6899
-
-
-
30
-
-
84926541865
-
-
To avoid confusion with the number n of spin components, we will use p to denote the number of replicas.
-
-
-
-
33
-
-
84926539910
-
-
The possibility of failure of dimensional reduction caused by many extrema of the Hamiltonian has been suggested by many workers independently;
-
-
-
-
39
-
-
84926579779
-
-
(private communication)has recently argued that if supersymmetry is broken, as it will be if there are many metastable states, then replica symmetry will also be broken. This is likely to occur due to the nonperturbative effects discussed in Sec. II.
-
-
-
Cardy, J.L.1
-
40
-
-
84926582590
-
-
In the Fourier variables Γm the flows in d=4+ epsilon are somewhat better behaved;
-
-
-
-
41
-
-
84926587783
-
-
however, generation of high-m anisotropies is still likely to cause problems with the uniformity of the expansion.
-
-
-
-
42
-
-
84926533437
-
-
The method used here for formally obtaining the supersymmetric nonlinear sigma model was shown to this author by J. L. Cardy, who had derived the supersymmetric result for the random-field case. The method originally used by this author was rather clumsy and less easily generalizable to include random anisotropy.
-
-
-
|
