We consider the conformal bootstrap for spacetime dimension

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2$. We will argue that this is evidence for a qualitative difference in the nature of the Ising model universality class below $d=2$, and will back this up with a detailed analysis of the spectra of solutions to crossing symmetry in the neighbourhood of the hypothetical Ising model for $d\lesssim 2$.
There are multiple reasons why the bootstrap in $1

2$, the Wilson-Fisher fixed point lies on a kink. It is interesting then to consider what happens to this kink for $d<2$. However, we immediately run into a puzzle: a closer look at our curves seems to show that there is not one but two inflection points! They can be seen especially clearly in the bounds for dimensions $1.875$ and $1.65$, and seem to fade away below $d=1.5$. To clear up the situation we must examine the spectra of the solutions to crossing symmetry along the boundary of the bounds. Kinks in the bounds have previously~\cite{El-Showk2014} been shown to be related to rearrangements in the spectrum of solutions as we vary $\Delta_\sigma$. Therefore, instead of looking for features in the bound plots, we shall instead consider the spectra and look for such rearrangements there. We begin by considering what happens as $d$ is lowered below two. In figure~\ref{fig:specSmall} we show the spectra close to $\Delta_\sigma=1/8$ (which is the correct value for the critical Ising model in $d=2$) for $d=1.98$ and $d=2$. For $d=2$ we see that there are sharp operator rearrangements taking place in both the spin 0 and spin 2 sectors. These two rearrangements lie very close to each other, and indeed we have checked that as we increase the level of the truncation (i.e.\ as our bounds become stronger) they approach each other. For $d=1.98$ we see that these rearrangements are still present, but a significantly larger distance apart. \begin{figure} \centering \includegraphics[width=.49\columnwidth]{n11_spectrum_2.pdf} \hfill \includegraphics[width=.49\columnwidth]{n11_spectrum_198.pdf} ]]>

1.65$ for all theories we considered. We have included only a representative selection of fractal Ising models, but we have checked that other results in literature~\cite{monceau1998magnetic,Carmona1998} fall within this general area and none satisfy our bounds. \begin{figure} \centering \includegraphics[width=13cm]{fig5.pdf} ]]>

0$ it satisfies a fourth-order equation. On the other hand, the Casimir equation above implies: \begin{eqnarray} \tilde D_{\varepsilon} g_{\tiny{\Delta,L}}(x)&=&-(1+2\varepsilon)\, (1-x)x^2\, h_{\Delta,L}(x), \\ \tilde D_{\varepsilon}&\equiv& \frac 12(1-x)x^2\, \partial_x^2-(1+a+b+\varepsilon)\,x^2\,\partial_x-2\,ab\,x-c_2\,. \end{eqnarray} We are interested in the limit $d\to 1 \Rightarrow \varepsilon \to -\frac 12$. In this limit there are two distinct possibilities. Suppose first that $h_{\Delta,L}$ is finite in this limit. Then it follows that the conformal block at $z=\bar z$ satisfies a second order differential equation, namely $\tilde D_{-\frac 12}\, g_{\Delta,L}=0$. Now, we already know that this function generically satisfies a third or fourth order differential equation, so this can only be true if such an equation factorizes for $\varepsilon=-\frac 12$. We find this to be precisely so only for the cases $L=0$ and $L=1$. We have exact expressions for these blocks when $z=\bar z$, derived in~\cite{ElShowk-ml-2012ht}: \begin{align} G_{\Delta,0}(z) &= \left(\frac{z^{2}}{1-z}\right)^{\Delta/2} \, _3F_2\left({\textstyle\frac{\Delta}2,\frac{\Delta}2,\frac{\Delta }{2}-\varepsilon ;\frac{\Delta+1 }{2},\Delta -\varepsilon} ;\frac{z^2}{4 (z-1)}\right)\,, \label{eq:3F2-0} \\ G_{\Delta,1}(z) &=\frac{2-z }{2 z} \left(\frac{z^2}{1-z}\right)^{\frac{\Delta +1}{2}} \, _3F_2\left({\textstyle\frac{\Delta+1 }{2},\frac{\Delta+1 }{2},\frac{\Delta+1 }{2}-\varepsilon ;\frac{\Delta }{2}+1,\Delta -\varepsilon };\frac{z^2}{4 (z-1)}\right)\,. \label{eq:3F2-1} \end{align} Taking $d\to 1$, or equivalently $\varepsilon \to -1/2$ we get: \begin{eqnarray} \lim_{d\to 1}\, G_{\Delta,0}(z)=\,\lim_{d\to 1} G_{\Delta,1}(z)= z^\Delta \ _2 F_1(\Delta,\Delta,2\Delta;z) \end{eqnarray} so they are the same. A cross-check is that the values of the Casimir for $L=0$ and $L=1$ are the same when $d=1$. Notice however that for $d$ infinitesimally close to one these blocks have a smooth analytic continuation into the region $z\neq \bar z$ where they \emph{are} distinct. For higher spins instead $h_{\Delta,L}(x)$ diverges as $d\to 1$. Considering higher orders in the $y$ expansion, no new sources of divergences are introduced, so that all other terms are either finite or divergent as $1/(d-1)$. This divergence suggests we should change the normalization of the $L\geq 2$ conformal blocks by the same factor - or equivalently, absorbing it into the OPE coefficients. When we do so, we see that the purely $z=\bar z$ piece of the block decouples as $d\to 1$ --- the blocks become purely transverse! This means that accordingly the bootstrap equations develop a decoupled sector consisting of the $L=0,1$ blocks at $z=\bar z$. Altogether, these results imply that for a generic four point function the crossing equations take the schematic form in the limit $d\to 1$: \begin{equation} \sum_{L=0} \lambda_{\Delta,0}^2 \left( \begin{tabular}{c} $F_{\Delta,0}^{\|}$\\ $F_{\Delta,0}^{\perp}$ \end{tabular} \right) + \sum_{L=1} \lambda_{\Delta,1}^2 \left( \begin{tabular}{c} $F_{\Delta,0}^{\|}$\\ $F_{\Delta,1}^{\perp}$ \end{tabular} \right) + \sum_{L>0} \hat \lambda_{\Delta,L}^2 \left( \begin{tabular}{c} $0$\\ $(d-1)F_{\Delta,L}^{\perp}$ \end{tabular} \right)= \left( \begin{tabular}{c} $0$\\ $0$ \end{tabular} \right). \end{equation} In the crossing symmetry relations for four identical scalars we must of course have $\lambda^2_{\Delta,L}=0$ for all odd spins $L$. The equations above shows that the crossing equations split into two parts. Firstly, the parallel equations --- denoted with $\|$ --- involve only scalars/spin-1 blocks with $z=\bar z$ and so form a decoupled sector. This decoupled sector is of course simply the purely $d=1$ bootstrap. Hence solutions to crossing symmetry for $d$ arbitrarily close to one have to at least satisfy the same constraints as those in $d=1$. Once these constraints are solved, the spin-0/spin-1 spectrum is completely fixed. Next step is to satisfy the remaining equations, which can be thought of as determining whether an analytic continuation into transverse space --- denoted by $\perp$ --- of this $d=1$ solution can exist. Indeed, since for $d>1$ we switch on the transverse parts of the spin-0 and spin-1 blocks, we will also need to turn on higher spins to cancel those. If we can solve these extra equations, then the analytic continuation exists and will be a smooth function of $d$. If it doesn't, then we are necessarily faced with a discontinuity in the bounds at $d=1$. In other words, the discontinuity can arise because the transverse parts of the $L=0,1$ blocks and also higher spins are by definition only turned on for $d$ strictly greater than one --- this adds a whole new set of crossing constraints which may not have a solution. ]]>

1$. These bounds are evaluated with the constraints truncated at only 10 components ($n_{\rm max} = 3$), but the discontinuity moves further from $d=1$ as the number of components increases. ]]>

1$ extension is to simply drop the $d=1$ constraint relating the $v$ to the $u$ cross-ratio. But this cannot work, since the expansion of such a four point function will include only odd-spin conformal blocks, whereas we are assuming that the $\sigma$ field is bosonic, and hence the correlation function decomposition should only include even spins. How is it then that nevertheless we obtain~(\ref{gff}) in $d=1$? The answer comes from the analysis of the previous section, where we showed that the $z=\bar z$ parts of the scalar and spin-1 blocks actually match in the $d\to 1$ limit. Hence, we may interpret the discontinuity as coming from the fact that $d=1$ is special in allowing us access to fermionic correlation functions using scalar conformal blocks. We expect then that repeating the bootstrap computations for $d>1$ allowing odd instead of even spin conformal blocks we should see a continuous limit --- a result which we have confirmed. It is interesting to compare these results with models in the literature. The planar interface model of Wallace and Zia discusses the dynamics of a codimension one defect separating two different phases of a $d$-dimensional thermodynamical system~\cite{Wallace1979}. The model is a simple DBI action for a $(d-1)$-dimensional brane which is expanded about an infinitely extended surface. One finds there is a weakly coupled UV fixed point for small $\epsilon=d-1$, with critical exponents known up to four loops~\cite{Forster1981}. It will be sufficient for our purposes to quote the leading result \begin{eqnarray} \nu=\frac{1}{\epsilon}+\mathcal O(1)\,. \end{eqnarray} Recall that $\nu$ is associated with the divergence of the correlation length with as $T\to T_c$. It determines the dimension of $\varepsilon$ in the Ising model, via \begin{eqnarray} \Delta_{\varepsilon}=d-\frac{1}\nu\simeq 1+\mathcal O(\epsilon^2)\,. \end{eqnarray} Similarly, the droplet model~\cite{Bruce1981} considers the configuration energy of surface tension of spherical droplets. Near $d=1$ the droplet distribution function can be computed exactly, and from this the magnetization. At the fixed point we can read off the $\eta$ critical exponent: \begin{eqnarray} d+\eta-2=\frac{8}{\pi}(\epsilon)^{-1-\epsilon/2} e^{-1-2 C-2/\epsilon} \end{eqnarray} with $C\simeq 0.577$ the Euler constant. The conformal dimension of the field $\sigma$ is then \begin{eqnarray} \Delta_{\sigma}=\frac 12 (d-2+\eta)=\frac{4}{\pi}(\epsilon)^{-1-\epsilon/2} e^{-1-2 C-2/\epsilon}\,. \end{eqnarray} It is important to note, that \emph{a priori} there is no fundamental reason for claiming that these critical exponents are those of the Ising model --- indeed there are arguments that these models do not fully capture the interface free energy of the Ising model~\cite{Huse1985} --- so this remains a conjecture. Nevertheless we can take this model as face value and try to find it in our plots. As should be clear however, the discontinuity in our bounds implies that there is no hope of matching the theoretical analysis of the droplet model, since they predict $\ds\to 0$ and $\Delta_{\varepsilon}\to 1$ when $d\to 1$. Clearly all such models are then ruled out as unitary conformally invariant fixed points. ]]>

2$. It would be surprising if it would stop being so for $d\lesssim 2$. The reason it does work for $d>2$ is that such non-unitarity shows up only at relatively high values of the conformal dimension of operators, and so this has a relatively small effect on say the dimension of the first scalar in the $\sigma \times \sigma$ OPE. However such operators are expected to appear at lower dimensions for $d<2$. In any case, we might naively think that continuity would imply that our results should match at least the Borel-resummed $\epsilon$-expansion for $d$ close to two. The fact that they do not suggests something fundamentally new is happening. We have argued that the bootstrap itself gives qualitative evidence for this, through the fact that the single kink present above $d=2$ surprisingly splits into two below it. In the context of $\lambda \phi^4$ theory, we also expect that indeed $d<2$ is different, since the UV of the renormalization group flow down to the hypothetical Wilson-Fisher fixed point is a non-unitary free theory. Accordingly $\phi^4$ acts as a perturbation with negative conformal dimension, which is unusual to say the least. Perhaps this might ultimately be the explanation for our results, but at this point it remains speculative. Our bounds do not provide much clarity to the complex collection of results surrounding the Ising model on fractal lattices. Of course there is no reason \emph{a priori} to expect any of these (non-translationally invariant) theories to lie within our bounds, although they do exhibit critical behaviour. It would be interesting to show that fractal Ising models come closer to our bounds as lacunarity decreases, however there are currently not enough numerical results for a study of this type to be pursued. And, given that the Wilson-Fisher fixed point appears to no longer be a unitary CFT in this regime it is entirely possible that a decrease in lacunarity will have no impact on the connection between fractal Ising models and our bounds. Another avenue for future research would be to consider fractal lattices generated by a random deletion of subsquares, which restores an average translational invariance in the large $k$ limit. Perhaps this, or some other topology, might produce a fractal Ising model corresponding to a unitary CFT in noninteger dimensions, but at this point it does not look too promising. Our results for $d\to 1$ are rather interesting. On very general grounds we have excluded models which give $(\Delta_\sigma,\Delta_{\varepsilon})\to (0,1)$ as $d\to 1$. This is precisely the behaviour expected for the Ising model in this limit --- e.g.\ the spin field correlation function becomes a constant close to the $T=0$ ``critical point'' in $d=1$. Hence this is perhaps the strongest evidence for our claim that the Wilson-Fisher fixed point is very different below $d<2$. Let us consider the nature of the solution to crossing symmetry obtained by setting ourselves at the bound and for $\Delta_\sigma$ sufficiently large such that we are above the kink. Due to the peculiar $d\to 1$ limit, the scalar sector and the higher spin sector decouple, and furthermore the $z=\bar z$ piece of the solution is given solely in terms of the scalar blocks. This piece is reproduced by the four point function of a generalized free fermion. It would be extremely interesting to determine what the solution looks like for $z\neq \bar z$. This would require understanding the conformal blocks in this limit analytically, which unfortunately seems difficult. Regarding the kink itself, we see that in spite of the very sharp change in the properties of the spectrum, no operator rearrangements are visible. In particular we see no analog of ``null states'' as in previous studies~\cite{El-Showk2014}. This suggests that this transition may be kinematical in nature, and not due to the existence of an interesting conformal field theory at this point. Overall, we may say the results in this note are a double edged sword: on the one hand, they exemplify the power of the conformal bootstrap in cutting through large swaths of conformal theory space and eliminating hypothetical fixed points; but on the other hand they remind us that there may be interesting critical systems which it cannot capture --- if there is indeed a non-unitary Wilson-Fisher fixed point for $d<2$, we will never see it. This is because the bootstrap approach depends on positivity in a crucial way, and this in turn follows from unitarity. In this respect, it would be interesting to pursue the approach promoted by Gliozzi~\cite{Gliozzi2014,Gliozzi2013} --- although it is less developed and understood, at least for such systems it might be superior, since it does not depend on unitarity. \newpage ]]>