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Solving the 3d Ising Model with the Conformal Bootstrap II. \(c\)-Minimization and Precise Critical Exponents

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Abstract

We use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge \(c\) in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several \(\mathbb {Z}_2\)-even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension \(\Delta _\sigma = 0.518154(15)\), and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.

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Notes

  1. These conditions are the obvious ones implied by conformal symmetry, unitarity, and crossing symmetry of \(\langle \sigma \sigma \sigma \sigma \rangle \). We expect that \(p_{\Delta ,\ell }\)’s in actual CFTs satisfy further conditions related to consistency of other four-point functions, and perhaps more exotic conditions like consistency of the theory on compact manifolds. However, we do not impose these constraints in defining \(\mathcal {C}_{\Delta _\sigma }\). It will be interesting and important to explore them in future work, for instance [5].

  2. Such a scalar \(\sigma \) is sometimes called a generalized free field.

  3. MFT is not a genuine local CFT because it doesn’t contain a stress-tensor. However, it does appear as a point in \(\mathcal {C}_{\Delta _\sigma }\).

  4. We give a full description of our algorithm in Sect. 6.2.

  5. On the other hand, we will not have to restrict the dimensions \(\Delta \) to a discrete set, as was done originally in [8]. For this work, we have implemented an optimization algorithm (described in Sect. 6.2) that works with continuously varying \(\Delta \).

  6. This equation is schematic. In Sect. 6.2.3 we will pass from \(u,v\) to the variables \(z,\bar{z}\), and take \(N\) partial derivatives with respect to those variables.

  7. The 3d Ising CFT can be obtained as an IR fixed point of the \(\phi ^4\) theory. The UV stress tensor then naturally gives rise to the IR stress tensor, but there is no reason to expect that a second operator with the same quantum numbers will emerge in the IR. This definitely does not happen in the \(\epsilon \)-expansion.

  8. In our previous work [1, 3] \(c\) was denoted \(C_T\).

  9. By contrast, bounds on \(c\) in 4d do not show a similar minimum [1012].

  10. We are aware of the fact that \(c\) is not always monotonic under RG-flow, e.g. [13].

  11. Because \(\Delta \) can vary continuously, \(p_T\)-maximization should more properly be called a semi-infinite program, although we will not be careful about this distinction.

  12. In the analysis of CFTs with global symmetry [12], up to 66 constraints were used per each of the three (for \(SO(N)\)) or six (for \(SU(N)\)) bootstrap equations present in that case. In [4], up to 55 constraints were used per bootstrap equation. See Appendix for a discussion of the methods used in these works.

  13. Here and in subsequent plots \(N=231\) data cover a smaller subinterval \(\Delta _\sigma \in [0.5180,0.5183]\).

  14. And even earlier in the two-dimensional case in [14].

  15. As well as an analogous bound on the dimension of the subleading spin 2 operator.

  16. This is sometimes called the Callan-Gross relation since it was first noticed in perturbation theory in [20].

  17. Another such class are the amplitude ratios. These are related to IR-dominated properties of RG flows produced when the theory is perturbed away from the critical point. Unlike the critical exponents, the amplitude ratios cannot be easily computed in terms of the CFT parameters.

  18. Henkel refers to \(|-\rangle \) and \(|+\rangle \) as \(|\sigma \rangle \) and \(|\epsilon \rangle \), but as will see below this notation may lead to a confusion in his geometry.

  19. An earlier reference [34] measured \(\Delta _\sigma \) and \(\Delta _\epsilon \) approximating \(S^2\) by a discretized cube.

  20. See also [40] for a study based on the functional renormalization group.

  21. \(\Delta ^{\text {max}}_\epsilon =1.000003\) for \(N=60\) [2].

  22. The OPE coefficients were obtained by expanding the exactly known four-point function \(\langle \sigma \sigma \sigma \sigma \rangle \) into \(SL(2,\mathbb {C})\) conformal blocks. When there are several quasiprimaries with the same dimension and spin, they are lumped together by summing their OPE coefficients squared.

  23. The first study of the \(c\) lower bound in two dimensions was done in [42], Sect. 6.3.2. A peculiarity of the 2d case is that for \(\Delta _\epsilon ^{\text {cutoff}}\) close to the unitarity bound, the lower bound on \(c\) disappears altogether, allowing solutions to crossing with an arbitrarily small \(c\).

  24. A state of dimension \(\sim \!\!6.5\) present only in a small interval around \(\Delta _\sigma =0.128\) and with a tiny OPE coefficient is clearly a numerical artifact.

  25. In the range \(\frac{1}{2} \le \Delta _\sigma \le 1\), the function \(g^{(\Delta _\sigma )}\) may still saturate the bounds on \(c\) and \(\Delta _\epsilon \), but this range has not been sufficiently explored and the conclusive numerical evidence is lacking.

  26. For the conformal blocks corresponding to identical external scalars used here, the expansion is actually in powers of \(r^2\) [46].

  27. We used this feature in our \(d=2\) computations presented in Sect. 5. Instead of dealing with double poles, we used our generic simple-pole code and ran it at \(d=2+10^{-5}\). We checked that using \(d=2+10^{-7}\) or even \(d=2+10^{-15}\) does not change the results.

  28. At non-generic vertices, it’s possible that extra inequalities can be saturated, so that some of the \(x_{j_i}\) actually vanish.

  29. Except possibly at one intermediate step after the hot start, see Sect. 6.2.5.

  30. The simplex algorithm can proceed as long as we always find a negative reduced cost at every step. Thus we can rescale \(\mathrm {RC}_{\Delta ,\ell }\) by any positive function of \(\Delta \), and we will still eventually find the correct optimum. However, different rescalings cause the search to proceed in different ways. For example, if we strip off the factor \(r_*^\Delta \) from \(\mathrm {RC}_{\Delta ,\ell }\), the simplex algorithm will proceed along a different path, favoring larger values of \(\Delta \) in the intermediate steps. By the exponential decoupling theorem of [9], low-lying operators will dominate the constraints of crossing symmetry, and it is more practical to start the search by exploring the low dimensions first. We have found that the normalization in Eq. (6.36) realizes this requirement in practice, and leads to the fastest solution times. This is because this normalization is natural from the point of view of the OPE convergence estimates of [9].

  31. An alternative way to avoid discretizing \(\Delta \) is to use semidefinite programming [4, 12], see Appendix.

  32. This truncation of spins is an approximation. By the unitarity bound, operators with large spin have large dimension, and one can show that the contribution of large dimension operators to a four-point function is exponentially suppressed at \(z=\bar{z} = 1/2\) [9]. However, if we investigate the constraints of crossing symmetry near the lightcone (\(z\rightarrow 0\) with \(\bar{z}\) fixed), then it is known that large spin operators play an important role [18, 19]. It will be interesting to explore this regime in future bootstrap studies.

  33. When the simplex algorithm terminates, we do want to be sure of the true minimum. In this case, we can either apply a more rigorous minimization strategy or simply decrease the tolerance parameter \(\epsilon \) in Eq. (6.38).

  34. Each time we split an interval, we can re-use our calculation of \(f'\) at the endpoints and midpoint in the next level of recursion.

  35. In fact, we use a hybrid of Newton’s method and binary search which is guaranteed to stay within the interval and find a zero.

  36. The algorithm actually does not terminate after a finite number of steps but instead converges quickly to a solution. One can terminate the algorithm by hand when the minimum reduced cost is sufficiently close to zero, \(\mathrm {min}(\mathrm {RC}_{\Delta ,\ell })\ge -\delta \) for small \(\delta \). Some of our plots (Figs. 1 (right), 4 (right), 5, 10, and 14) were produced with \(\delta =10^{-60}\). We have found that a much less conservative criterion \(\delta \lesssim 10^{-12}\) suffices to reach the optimal spectrum with reasonable precision. Another criterion is to terminate if \(p_T\) is reduced by less than \(\delta \) in the last \(M\) iterations (we used \(M=1,000\), \(\delta =10^{-15}\)). The point is that the minimum reduced cost is not a perfect predictor for the actual reduction in \(p_T\), and it may also fluctuate significantly from one step to the other. Thus looking at the change in \(p_T\) integrated over many iterations may give a better idea about the progress of the algorithm.

  37. These nearby pairs of dimensions are responsible for making the matrix \(\mathbf {A}_B\) nearly-degenerate. In technical language, its condition number decreases as the algorithm progresses. This is a well-known potential numerical instability of the semi-infinite programming problems, see [49], Fig. 3. It is one of the reasons why we have to work with the multiple precision arithmetic—to avoid large rounding errors when inverting the poorly-conditioned matrix \(\mathbf {A}_B\). Another reason for using multiple precision is not related to the “spectrum doubling”—it comes from the fact that for large \(N\) there is a huge disparity in size between low and high-order derivative components in the vectors \(\mathbf {F}_{\Delta ,\ell }^{\Delta _\sigma }\) composing the matrix \(\mathbf {A}\). Computing the minimum reduced cost, Eq. (6.35), is an arithmetic operation which mixes these components. Since the minimum reduced cost may become tiny at the final steps of the algorithm, we have to perform its computation at a sufficient number of digits to be able to determine it accurately.

  38. We use Cython [50] to link the MPFR multiple precision arithmetic library [51], and to compile more computationally intensive parts of our Python code.

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Acknowledgments

We are grateful to M. Hasenbusch, M. Henkel, D. Mouhanna and E. Vicari for the useful communications concerning their work. We are grateful to B. van Rees for the discussions of the interpolating solution. In addition, we thank N. Arkani-Hamed, C. Beem, A. L. Fitzpatrick, G. Fleming, H. Ooguri, H. Osborn, J. Kaplan, E. Katz, F. Kos, J. Maldacena, J. Penedones, L. Rastelli, N. Seiberg, and A. Zhiboedov for related discussions. S. R. is grateful to the Samara Chernorechenskaya Scientific Center for their hospitality. S. R., D. S. D., and D. P. are grateful to KITP for their hospitality. We would also like to thank the organizers and participants of the Back to the Bootstrap 3 conference at CERN. This research was supported in part by the National Science Foundation under Grant No. PHY11-25915. The work of S. E. was partially supported by the French ANR contract 05-BLAN-NT09-573739, the ERC Advanced Grant no. 226371 and the ITN programme PITN-GA-2009-237920. M.P. is supported by DOE Grant DE-FG02-11ER41742. A. V is supported by DOE Grant DE-AC02-05CH1123. The work of D. S. D. is supported by DOE Grant number DE-SC0009988. Computations for this paper were run on National Energy Research Scientific Computing Center supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123; on the CERN cluster; on the Aurora and Hyperion clusters supported by the School of Natural Sciences Computing Staff at the Institute for Advanced Study; on the Omega cluster supported by the facilities and staff of the Yale University Faculty of Arts and Sciences High Performance Computing Center; on the TED cluster of the Chemistry Department and High Energy Theory group at Brown University; and the Kelvin cluster at the C. E. A. Saclay funded by the European Research Council Advanced Investigator Grant ERCAdG228301. S. E. would like to thank D. Kosower for providing access to the Kelvin cluster.

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Appendix: Comparison to Semidefinite Programming

Appendix: Comparison to Semidefinite Programming

An alternative optimization algorithm that has proved useful in bootstrap studies is semidefinite programming [4, 12]. Like our algorithm presented here, semidefinite programming avoids discretizing the possible operator dimensions \(\Delta \). The starting point is an approximation for linear functionals in terms of polynomials times positive functions,

$$\begin{aligned} \left. \sum _{m,n}a_{mn}\partial ^m_r \partial ^n_\eta G_{\Delta ,\ell }(r,\eta )\right| _{r=r_*,\eta =1} \approx \frac{r_*^\Delta }{Q_\ell (\Delta )}\sum _{m,n}a_{mn}P^{m,n}_\ell (\Delta ), \end{aligned}$$
(8.1)

where \(P_{\ell }^{m,n}(\Delta )\) are polynomials and \(Q_\ell (\Delta )=\prod _i (\Delta -\Delta _i)\). This approximation follows from Eq. (6.5). Positivity of the polynomial \(\sum _{m,n}a_{mn}P^{m,n}_\ell (\Delta )\) can be encoded in terms of \(r\times r\) positive semidefinite matrices, where

$$\begin{aligned} r=\max _{m,n}\left\lceil \frac{1}{2} \deg P^{m,n}_\ell (\Delta )\right\rceil =\left\lceil K+\frac{1}{2} q\right\rceil . \end{aligned}$$
(8.2)

Here, the number of derivatives satisfies \(m+n\le 2K\), as in Sect. 6.2.3, and \(q\) is the number of poles \(\Delta _i\) included in the rational approximation (8.1).

The most popular semidefinite program solvers used in the bootstrap are SDPA and its arbitrary precision version SDPA-GMP [54]. Their performance scales differently from the performance of our algorithm. In practice, the running time increases slowly as the number of crossing relations \(n_C\) is increased, but quickly as the degrees of the polynomial approximations \(r\) (\(K\) or \(q\)) are increased. (By contrast, our algorithm is relatively insensitive to the number of poles \(q\).) This makes SDPA a good choice for studying theories with global symmetries, where we have different crossing relations for each tensor structure which can appear in a four-point function [55]. (For example, theories with \(O(N)\) symmetry have \(n_C=3\) for a four-point function of vectors. Theories with \(SU(N)\) symmetry have \(n_C=6\) for a four-point function of anti-/fundamentals.) The overall dimension of the space of linear functionals is

$$\begin{aligned} d_\Lambda = n_CN = n_C\frac{K(K+1)}{2}. \end{aligned}$$
(8.3)

In [12], this was taken as high as \(n_C=6\) and \(K=11\), so \(d_\Lambda =396\). Each optimization for a problem of this size takes approximately 48 h.

In this work, we have been interested in studying a single crossing relation \(n_C=1\) and exploring as many derivatives as possible. We have been able to reach \(K=21\), so that \(d_\Lambda =231\). Although this value of \(d_\Lambda \) is smaller than what was achieved with SDPA-GMP in the case of global symmetries, the number of derivatives \(K=21\) could be difficult to match with SDPA-GMP. As the number of derivatives \(K\) is increased, one must also increase the number of poles \(q\) to maintain numerical stability. Further, one should tune the SDPA-GMP parameters to ensure the solver uses good initial data and termination criteria. It will be important to explore whether this can be done in the future.

SDPA-GMP uses a primal-dual solution method, so in principle it could be used for precision spectrum studies similar to what we do here. This will be interesting to explore in future work.

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El-Showk, S., Paulos, M.F., Poland, D. et al. Solving the 3d Ising Model with the Conformal Bootstrap II. \(c\)-Minimization and Precise Critical Exponents. J Stat Phys 157, 869–914 (2014). https://doi.org/10.1007/s10955-014-1042-7

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