Fractal Dimensions of Self-Avoiding Walks and Ising High-Temperature Graphs in 3D Conformal Bootstrap

Abstract

The fractal dimensions of polymer chains and high-temperature graphs in the Ising model both in three dimension are determined using the conformal bootstrap applied for the continuation of the O(N) models from $N=1$ (Ising model) to $N=0$ (polymer). Even for non-integer N, the O(N) sum rule allows one to study the unitarity bound formally defined from the positivity, which may be violated in a non-unitary CFT. This unitarity bound of the scaling dimension for the O(N)-symmetric-tensor develops a kink as a function of the fundamental field as in the case of the energy operator dimension in the $\mathbb {Z}_2$ (Ising) sum rule. Although this kink structure becomes less pronounced as N tends to zero, we found instead an emerging asymmetric minimum in the current central charge $C_J$. Despite the non-unitarity of the O(N) model at non-integer N, we find the $C_J$-kink along the unitarity bound lies very close to the location of the infrared (IR) O(N) CFT estimated by other methods. It is pointed out that certain level degeneracies at the IR CFT should induce these singular shapes of the unitarity bounds. As an application to the quantum and classical spin systems, we also predict critical exponents associated with the $\mathcal {N}=1$ supersymmetry, which could be relevant for locating the corresponding fixed point in the phase diagram.

Computations on the N-derivatives by the Fixed Dimension RG

We use the fixed dimension RG ($D=3$) augmented by the pseudo-$\epsilon$ series [89] to circumvent the accumulation of the intermediate systematic error due to the determination of the coupling $g^*$ at the non-trivial fixed point. The beta function is generalized by a new parameter $\tau$ such that at $\tau =1$ it reduces to the original beta function $\beta (g)$ in 3D:

$$\begin{aligned} \beta (g,\tau )= - \tau g + \beta _2(g), \end{aligned}$$

(49)

where $\beta _2(g)\equiv \beta (g)+g$ starts at order $g^2$ with a positive coefficient of order 1. Then the critical exponents can be expanded in $\tau$ by eliminating g by using $g=g^*(\tau )$ which solves $\beta (g^*,\tau )=0$. We compute the $\tau$-series for the derivatives of $\Delta _S$ and $\Delta _T$ at the degeneration point ($N=0$) based on the six-loop 3D RG results for $\beta (g)$, $\eta$, $\gamma ^{-1}$ [41] and $\eta _T=\eta +\phi _2/\nu -2$ [69]. The results are,²¹

$$\begin{aligned} \left. \frac{\partial }{\partial N}\right| _{N=0}\Delta _S=&\frac{3}{32}\tau +0.0241609 \tau ^2+0.0036762 \tau ^3\nonumber \\&+0.0021794 \tau ^4+0.0004817 \tau ^5-0.0019009 \tau ^6+\mathcal {O}\left( \tau ^7\right) \nonumber \\ =&\frac{3}{32} \epsilon +0.0361328 \epsilon ^2-0.0198967 \epsilon ^3+0.0380668 \epsilon ^4-0.0611648 \epsilon ^5+\mathcal {O}\left( \epsilon ^6\right) ,\end{aligned}$$

(50)

$$\begin{aligned} \left. \frac{\partial }{\partial N}\right| _{N=0}\Delta _T=&-\frac{1 }{32}\tau -0.0099826 \tau ^2 +0.0049121 \tau ^3\nonumber \\&\quad -0.0029128 \tau ^4+0.0066431 \tau ^5-0.0101052 \tau ^6+\mathcal {O}\left( \tau ^7\right) \nonumber \\ =&-\frac{1}{32} \epsilon -0.0146484\epsilon ^2+0.032921\epsilon ^3+0.028875\epsilon ^4+ \mathcal {O}\left( \epsilon ^5\right) , \end{aligned}$$

(51)

where we also show the derivatives computed from $\epsilon =4-d$ expansion up to known orders [91, 92] just for comparison. Since both $\tau$-series do not show strong asymptotic behaviors with factorial growth of coefficients up to the orders presented, even the naive direct summation of the series would be of some use; in particular, it is clearly better than the direct sum of the $\epsilon$-expansion.

Table 5 The Padé table for the derivative $\partial \Delta _S(0)/\partial N$

Table 6 The Padé table for $\partial \Delta _T(0)/\partial N$

A simple Padé analysis, however, may improve the stability of analysis as usual. This can be irrustrated as follows. We show the values from the Padé approximants [M / L] for these derivatives in Tables 5 and 6, respectively. The positive real poles closest to $\tau =1$ are shown in brackets for the six-loops (anti-diagonals $L+M=6$) and five-loops ($L+M=5$) order approximants. For each derivative, the data occuring with a pole in [0.5, 1.5] (indicated by $^*$) is omitted since it is rather close to $\tau =1$, where the series is to be evaluated. As a simple estimate, we take the average of the six and five-loops and the maximum deviation as an error. This gives the value quoted in (10) and (11) in the text.