Particle approach to loop models

This is based on the 2020 article by Delfino, which is partly a review article. The particle approach is able to recover exact results on the O(N) and Potts models, which were originally derived in the Coulomb gas approach. Moreover, the approach is applicable to disordered systems, and might lead to new exact results on the corresponding CFTs. (Section 2 of the article is a nice summary on critical phenomena.)

Case of the O(N) model
The idea is to have particles labelled by $$a=1,\dots, N$$, and to consider two-particle states $$|ab\rangle$$. Then $$O(N)$$ symmetry constrains scattering to take the form (51):

|ab\rangle \to \delta_{ab} S_1 \sum_{c=1}^N |cc\rangle + S_2 |ba\rangle +S_3 |ab\rangle $$ (Compared to the article, we do the switch $$S_2\leftrightarrow S_3$$. This is more consistent with Figure 4, and makes some formulas simpler.) This encodes an $$S$$-matrix acting on two-particle states,

S_{ab}^{cd} = S_1 \delta_{ab}\delta^{cd} +S_2\delta_a^d\delta_b^c + S_3 \delta_a^c \delta_b^d $$ (See Figure 4 for the graphical interpretation. In particular, $$S_2$$ is associated to loops that cross.) We have to impose the constraints of unitarity (49) and crossing (50). These constraints a priori use complex conjugation, and therefore break analyticity. However, crossing $$S_{ab}^{cd} = (S_{ac}^{bd})^*$$ (which amounts to $$S_1^*=S_3$$ and $$S_2^*=S_2$$) allows us to rewrite unitarity $$SS^* = 1$$ in an analytic way, which reduces to

S_1S_3+S_2^2 =1 \quad, \quad S_2(S_1+S_3)=0 \quad , \quad NS_1S_3+S_1^2+S_3^2 = 0 $$ (This calculation can be done graphically.) We are interested in solutions such that $$S_2=0$$, which implies $$N=-S_1^2 - S_1^{-2}$$. Then the singlet $$|v\rangle = \sum_a |aa\rangle$$ scatters as $$|v\rangle \to (NS_1+S_2+S_3)|v\rangle= -S_1^3|v\rangle$$.

Now, the claim is that the scattering phase for the singlet is $$e^{-2\pi i\Delta_{(2, 1)}}$$ (48), with $$\Delta_{(2,1)}=-\frac12 + \frac34\beta^2$$. This leads to $$S_1^3=e^{\frac32 \pi i\beta^2}$$. This is obtained by assuming that the particles are created by chiral fields (since they are massless) of spin $$\Delta_{(2,1)}$$, and the antiparticles by chiral fields of spin $$-\Delta_{(2,1)}$$: the phase corresponds to a rotation by $$\pi$$, times the difference of these spins. $$\Delta_{(2,1)}$$ is determined by mutual locality with the energy field $$V_{(1,3)}$$: we have $$\begin{aligned} V_{(1,3)\overline{(1,3)}} \times V_{(2,1)\overline{(1,1)}} = V_{(2,3)\overline{(1,3)}}\end{aligned}$$ and mutual locality, i.e. the fact that spins differ by integers, is guaranteed by the identities

\Delta_{(2,1)} = -\frac12 +\frac34\beta^2 \quad, \quad \Delta_{(1,3)} = -1+2\beta^{-2} \quad , \quad \Delta_{(2,3)} = -\frac52+\frac34\beta^2 +2\beta^{-2} $$ so that $$\Delta_{(2,3)} -\Delta_{(1,3)} = \Delta_{(2,1)}-1$$. More generally, we have

\Delta_{(r,3)} -\Delta_{(1,3)} = \Delta_{(r,1)}- (r-1) $$ so that any degenerate chiral field of dimension $$\Delta_{(r,1)}$$ with $$r\in\mathbb{N}^*$$ is mutually local with $$V_{(1,3)}$$. The case $$r=2$$ is only the lowest-dimensional non-trivial case.

Other derivation of the relation between $$c$$ and $$N$$: straightforward if we know the loop weight $$w = 2\cos(2\pi \beta P)$$, then $$N=w(P_{(1,1)})$$ corresponds to the identity field. But why would the weight of an oriented loop be $$e^{2\pi i\beta P}$$? Hard to escape Coulomb gas considerations.

These considerations generalize to the Potts model.

Is this related to the topological defects in the O(N) model, whose endpoints are fields of dimension $$\Delta_{(2,1)}$$?

Disordered O(N) model
This is obtained by taking $$n$$ replicas, and then doing $$n\to 0$$. Adding replicas is not the same as doing $$N\to nN$$, because the permutation symmetry between replicas allows couplings to depend on whether particles belong to the same replica or not. This leads to six possible interactions:

S_{a_ib_j}^{c_kd_l} = S_1 \delta_{ab}\delta^{cd}\delta_\text{all} +S_2\delta_a^d\delta_b^c\delta_\text{all} + S_3 \delta_a^c \delta_b^d\delta_\text{all} + S_4 \delta_{ab}\delta^{cd}\delta_{i=j\neq k=l} +S_5\delta_a^d\delta_b^c\delta_{i=l\neq j=k}  + S_6 \delta_a^c \delta_b^d\delta_{i=k\neq j=l} $$ where $$\delta_\text{all} = \delta_{i=j=k=l}$$. In the analytic unitarity equations, we have to distribute replica labels: the factor $$N$$ for a closed loop gets multiplied with $$1,n-2$$ or $$n-1$$, depending how many replica indices are allowed in that loop. Then the analytic unitarity conditions read

S_1S_3+S_2^2 = S_4S_6+S_5^2 =1 \quad ,\quad (S_1+S_3)S_2 = (S_4+S_6)S_5 = 0 $$

S_1^2+S_3^2+NS_1S_3+N(n-1)S_4S_6 = 0 $$

S_1S_4+S_3S_6 + N(S_1S_6+S_3S_4) + N(n-2)S_4S_6 +(S_4+S_6)S_2=0 $$ The case of non-intersecting loops is $$S_2=S_5=0$$. If in addition $$n=0$$, this leads to $$S_1^4=-1$$ and $$S_4=S_1$$ or $$S_4=S_1\frac{N-i}{N+i}$$, see Eq. (117).

See more detail in the article by Delfino and Lamsen. It is not clear if we can obtain as much information as in the (non-disordered) O(N) model about the central charge (as a function of $$N$$) or about conformal dimensions of a few fields.

In the case of the Potts model, Delfino's prediction of superuniversality, i.e. that the critical exponent $$\nu=\frac{1}{2-2\Delta_\text{energy}}$$ is one for any $$Q\geq 2$$, contradicts previous results by Dotsenko, Picco and Pujol (Eq. (4.17)), confirmed by Jacobsen.