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The determinant method in the conformal bootstrap is applied for the critical phenomena of a single polymer in arbitrary

The conformal field theory in arbitrary dimensions was developed a long time ago [

Instead of taking many relevant operators, the determinant method with small prime operators provides interesting results for the non-unitary cases. The determinant method is applied on the Yang–Lee edge singularity with considerable accuracy [

This paper deals with two different polymers using the determinant method: the single polymer, and branched polymers in a solvent. They have different upper critical dimensions, 4 and 8 respectively. It is well known that the polymer in a solvent is equivalent to a self-avoiding walk, which was studied by the renormalization group

A branched polymer in

We are concerned with two issues related to polymers: (i) the critical phenomena of polymers belong to the logarithmic conformal field theory since the central charge

In this paper we evaluate the scale dimensions of the single polymer and a branched polymer by the determinant method with a small number of operators. This study is an extension of a previous analysis of the Yang–Lee edge singularity [

Although we use these constraints in the determinant method, we extend the analysis by introducing a small difference between

The bootstrap method uses the crossing symmetry of the four-point amplitude. The four-point correlation function for the scalar field

The crossing symmetry of

In the previous paper [

There are many examples of critical phenomena that are non-unitary. A negative value of the coefficients of the operator product expansion (OPE) leads to the non-unitary case. For instance, this can be seen in the case of the Yang–Lee edge singularity and in polymers. For such non-unitary critical phenomena the unitarity condition does not hold, and direct application of the unitarity boundary condition does not work. For instance, an

On the other hand, the determinant method works for the non-unitary Yang–Lee edge singularity [

However, one needs to solve some difficulties in the replica limit

The catastrophic divergence is the factor

The second catastrophe, which is related to the central charge

We consider the case of the self-avoiding walk or polymer in a solvent. For this case, which corresponds to the

At the upper critical dimension

For three dimensions, the previous conformal bootstrap method gives the values of

In

The

Scale dimensions of a single polymer.

2 | 0.1 | 0.7 | 0.666 |

3 | 0.514 | 1.3 | 1.299 |

3.5 | 0.75 | 1.57 | 1.57 |

4 | 1.0 | 2.0 | 2 |

The central charge

There is a remarkable equivalence, the so-called dimensional reduction, between a branched polymer in

The action of the branched polymer has branching terms in addition to the self-avoiding term (single polymer). We write this action for the

The term

As before, the

The last term of 1 is a trivial term due to the definition of

In above formula, if we put

The exponent

This leads to the Yang–Lee edge singularity,

The condition

By dimensional reduction, the values of the exponents

In general dimension

We get the following relations:

In

Branched polymer in

We confirm the dimensional reduction to the Yang–Lee model in

In

Branched polymer in D=8: The zero loci of the

In

Branched polymer in

We have analyzed a single polymer and a branched polymer, and we have found that they are characterized by the degeneracies of the primary operators,

For a single polymer, the scaling dimension

We find in the branched polymer case the exact relation of

The validity of the dimensional reduction in a random field Ising model has long been discussed, and it is known that the reduction to a pure Ising model does not work in the lower dimensions. We will discuss this problem by the conformal bootstrap determinant method in a separate paper [

The author is grateful to Nando Gliozzi for discussions concerning the determinant method. He also thanks Edouard Brézin for useful discussions on the dimensional reduction problem in branched polymers. This work is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant-in-Aid 16K05491. This work is funded by Okinawa Institute of Science and Technology Graduate University.

Open Access funding: SCOAP