Chapter III: Bootstrapping Heisenberg Magnets and their Cubic Instability
3.4 Results
or in-progress points with∆gap ≤ ∆ceiling. At the beginning, there are no infeasible points at large gaps, so∆ceiling= ∆max_gap.
We return the center of the box at∆gap = ∆feasible+∆ceiling/2, thus bisecting the range of feasible gaps. This underscores the need for a good estimate of∆max_gap. If the estimate is too high, then the algorithm will recommend too many points that are far too large.
Overall, we have found this approach to work reasonably well. More importantly, it is very robust. It is very easy to be too clever, resulting in odd failures.
Figure 3.5: TheΛ = 43 conformal bootstrap dimension island (black) compared with the Monte Carlo results [7, 8] (green).
hull of the points that are midway between the allowed and disallowed vertices in these triangles. AtΛ= 43, this "best-fit" region gives
∆φ =0.518942(51∗),
∆s =1.59489(59∗),
∆t =1.20954(23∗). (3.20)
A more rigorous determination can be made by taking the convex hull of the disallowed points in these boundary Delaunay triangles. This region gives the rigorous error bars
∆φ = 0.518936(67), (3.21)
∆s = 1.59488(81), (3.22)
∆t = 1.20954(32), (3.23)
which we have quoted in table 3.1.
The allowed points atΛ=43 are associated with OPE coefficient ratios which live
0.5184 0.5186 0.5188 0.5190 0.5192 0.5194 0.5196 Δϕ 1.592
1.594 1.596 1.598 Δs
CB(Λ=19) CB(Λ=27) CB(Λ=35) CB(Λ=43) MC(HV '11) MC(H '20)
0.5184 0.5186 0.5188 0.5190 0.5192 0.5194 0.5196 Δϕ 1.208
1.209 1.210 1.211 Δt
CB(Λ=19) CB(Λ=27) CB(Λ=35) CB(Λ=43) MC(HV '11) MC(H '20)
1.592 1.594 1.596 1.598 1.600 Δs
1.208 1.209 1.210 1.211 Δt
CB(Λ=19) CB(Λ=27) CB(Λ=35) CB(Λ=43) MC(HV '11) MC(H '20)
Figure 3.6: Comparison between the conformal bootstrap islands at Λ = 19,27,35,43 projected to the{∆φ,∆s},{∆φ,∆t}, and{∆s,∆t}planes and the Monte Carlo results of [7, 8].
in the ranges11
λsss
λφφs =0.9643(20∗), λtts
λφφs =1.87593(53∗), λφφt
λφφs =1.66808(23∗), λttt
λφφs =2.86034(61∗). (3.24)
These should be viewed as an approximation to the full allowed region of OPE coefficients, which may be slightly larger.
Central charges andλφφs
Next, we compute upper and lower bounds on the magnitude of the OPE coefficient λφφs, the central chargeCT, and the current central chargeCJ. We compute these bounds over a small sample of points in our allowed region so the results will be inherently non-rigorous. However, we believe that this treatment gives reasonable estimates for these quantities that are more precise than previous results.
The strategy is similar to the method we employed in [4]. We take seven primal points in theΛ= 43 island, consisting of the scaling dimensions and allowed OPE coefficients. The points are chosen to be sufficiently symmetrized and sparse across theΛ = 43 island we have computed. For each of these points, we extremizeCT, CJ, and the external OPE norm parameterized by λφφs, to obtain upper and lower bounds. This calculation was limited toΛ=35 due to our available computational resources. The data points andSDPBparameters we used are summarized in tables B.5 and B.2, respectively.
There is an important comment we want to make about the upper bound computation onCT andCJ (a similar comment was made in [4]). For computing upper bounds onCT andCJ, we have to assume a gap∆T0/J0 above the unitarity bound for the next operators in theTorJsectors. Note that this gap was not assumed in our OPE scan, so this extra constraint might turn an allowed point into a disallowed point. If we do not have such a gap, the upper bound is loose and may not give reasonable results.
On the other hand, large gaps can make SDPB unable to find a solution.
11Note that there is an ambiguity in the signs of these coefficients, related to performing the operator redefinitionss → −sandt → −t. This freedom can be used to fixλφφs andλφφt to be positive, after which all other signs in (3.24) are determined to be positive by the conformal bootstrap.
In table B.5, we summarize the gaps∆T0/J0 we assume in the upper bound calcu- lations. From spectrum determinations using the extremal functional method (see [190, 191]), we have noticed that a gap∆T0/J0 =∆T/J+1 aboveT andJis generally favored. We were able to compute bounds with this gap for three of the points, but for the other four we could not find solutions. For those points, we adopted the weaker assumption∆ext,T(J) = ∆T/J +0.1.
Following this procedure, we obtain our estimates ofCT,CJ, andλφφsin the critical O(3)model,
CJ/CJfree =0.90632(16∗), CT/CTfree =0.944524(28∗),
λφφs = 0.524261(59∗). (3.25)
These results agree with and are more precise than previous determinations of these quantitites (see [162, 163, 178]).
Upper bound on∆t4
Our last result is the maximum value of the rank-4 scalar dimension ∆t4. In con- junction with the tiptop algorithm described in section 3.3, we computed points atΛ= 19,27,and 35. Allowed points at lower values ofΛwere used to initiate the search at larger values ofΛ. Figure 3.7 shows a projection of a subset of the 1311 disallowed points and 172 allowed points atΛ= 35, and Figure 3.8 shows how the island shrinks as we approach the maximum∆t4.
The largest allowed value of ∆t4 was 2.99052, with the tip centered around the scaling dimensions {∆φ,∆s,∆t} = {0.518962,1.59527,1.20969}. We can also consider the nearest disallowed point. To compute the nearest, we first use the affine transformation (B.8) which makes the dimension island roughly spherical withO(1) size, and we additionally rescale ∆t4 − 3 by a factor of ∼ 100 so that the tip’s curvature also ranges over anO(1)distance. Then we compute the Euclidean norm.
The algorithm is very well converged, so the result is robust against the precise form of these transformations. This gives us a conservative bound of
∆t4 <2.99056. (3.26)
This implies that the leading rank-4 tensor in the criticalO(3)model is relevant, in agreement with other studies.
Figure 3.7: Two-dimensional projection of the results of thetiptopsearch atΛ= 35. Thexcoordinate is related to the three scalar dimensions via (B.8). Projections in y and z look similar. We have superimposed a convex hull encompassing the allowed points on top, obscuring some of the disallowed points. We can see the behaviour of thetiptopalgorithm, exploring the island at one∆t4 before jumping to a larger ∆t4. The jumps become progressively smaller, indicating convergence.
We computed 16 points simultaneously, and this calculation took several months during which the tiptop algorithm was being developed. So the points reflect occasional crashes and small inefficiencies in the set of computed points.