Abstract
Abstract Dimensional reductions in the branched polymer model and the random field Ising model (RFIM) are discussed by a conformal bootstrap method. Small minors are applied for the evaluations of the scale dimensions of these two models and the results are compared to the $D'=D-2$D Yang–Lee edge singularity and to the pure $D'=D-2$D Ising model, respectively. For the former case, the dimensional reduction is shown to be valid for $3 \le D \le 8$ and, for the latter case, the deviation from the dimensional reduction can be seen below five dimensions.
Highlights
The critical exponent of a D-dimensional branched polymer, which is a polymer with trivalent branches in a D-dimensional solvent, is known to be same as the critical exponent of the D D Yang– Lee edge singularity with D = D − 2
The Yang–Lee edge singularity is a good example of the determinant method, which we will apply to random magnetic field Ising model (RFIM) later
It is known that the dimensional reduction works for (i) the branched polymer in D dimensions, which is equivalent to the Yang–Lee edge singularity in D = D − 2 dimensions, and (ii) the electron density of state in a 2D random impurity potential under a strong magnetic field [28,34]
Summary
4, the dimensional reduction of the branched polymer to the Yang–Lee edge singularity is explained by a supersymmetric argument similar to RFIM. 5, we discuss the dimensional reduction of the branched polymer to the Yang–Lee edge singularity by the determinant method. The Yang–Lee edge singularity is a good example of the determinant method, which we will apply to RFIM later It originates from the critical behavior of the density of the zeros of the partition function of the Ising model with a complex magnetic field. It is known that the dimensional reduction works for (i) the branched polymer in D dimensions, which is equivalent to the Yang–Lee edge singularity in D = D − 2 dimensions, and (ii) the electron density of state in a 2D random impurity potential under a strong magnetic field [28,34]. This may give a possible proof of the dimensional reduction from D to D − 2, but as we discussed before, this dimensional reduction does not work, since the measure does not show the positivity
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