Erratum to: Scaling dimensions of monopole operators in the ℂ ℙ N b − 1 $$ \mathrm{\mathbb{C}}{\mathrm{\mathbb{P}}}^{N_b-1} $$ theory in 2 + 1 dimensions

The scaling dimensions of charged operators in conformal field theory have recently been predicted to exhibit universal behavior in the large charge limit. We verify this behavior in the 2+1 dimensional CP N −1 model. Specifically, we numerically compute the scaling dimensions of the lowest dimension monopole operators with charges Q = 1, 2, · · · , 100 to subleading order in large N . The coefficients of the large Q expansion are extracted through a fit, and the predicted universal O(Q 0 ) contribution is verified to the subpercent level. arXiv:1805.00501v1 [hep-th] 1 May 2018

“…The CP N −1 model involves an N -component complex scalar field φ, a U (1) gauge field A µ , and a Lagrange multiplier field λ. The Lagrangian is [26][27][28][29][30][31]…”

Section: Cp N −1 Modelmentioning

confidence: 99%

Section: Saddle Point and Critical Couplingmentioning

confidence: 99%

“…. , 5 in a computational tour de force [31]. We simply repeated that computation for Q = 1, 2, · · · , 100.…”

Section: Computationmentioning

confidence: 99%

“…We simply repeated that computation for Q = 1, 2, · · · , 100. For all the details, see [31]. Note that our definition of Q differs from [31] by a factor of 2.…”

Section: Computationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The large charge expansion at large N

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Superfluids, vortices and spinning charged operators in 4d CFT

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Accessing large global charge via the ϵ-expansion

Watanabe

2021

We compute the lowest operator dimension ∆(J; D) at large global charge J in the O(2) Wilson-Fisher model in D = 4 − ϵ dimensions, to leading order in both 1/J and ϵ. While the effective field theory approach of [1] could only determine ∆(J; 3) as a series expansion in 1/J up to an undetermined constant in front of each term, this time we try to determine the coefficient in front of J3/2 in the ϵ-expansion. The final result for ∆(J; D) in the (resummed) ϵ-expansion, valid when J ≫ 1/ϵ ≫ 1, turns out to be$$ \Delta \left(J;D\right)=\left[\frac{2\left(D-1\right)}{3\left(D-2\right)}{\left(\frac{9\left(D-2\right)\pi }{5D}\right)}^{\frac{D}{2\left(D-1\right)}}{\left[\frac{5\Gamma \left(\frac{D}{2}\right)}{24{\pi}^2}\right]}^{\frac{1}{D-1}}{\epsilon}^{\frac{D-1}{2\left(D-1\right)}}\right]\times {J}^{\frac{D}{D-1}}+O\left({J}^{\frac{D-2}{D-1}}\right) $$ Δ J D = 2 D − 1 3 D − 2 9 D − 2 π 5 D D 2 D − 1 5 Γ D 2 24 π 2 1 D − 1 ϵ D − 1 2 D − 1 × J D D − 1 + O J D − 2 D − 1 where next-to-leading order onwards were not computed here due to technical cumbersomeness, despite there are no fundamental difficulties. We also compare the result at ϵ = 1,$$ \Delta (J)=0.293\times {J}^{3/2}+\cdots $$ Δ J = 0.293 × J 3 / 2 + ⋯ to the actual data from the Monte-Carlo simulation in three dimensions [2], and the discrepancy of the coefficient 0.293 from the numerics turned out to be 13%. Additionally, we also find a crossover of ∆(J; D) from ∆(J) ∝ $$ {J}^{\frac{D}{D-1}} $$ J D D − 1 to ∆(J) ∝ J, at around J ∼ 1/ϵ, as one decreases J while fixing ϵ (or vice versa), reflecting the fact that there are no interacting fixed-point at ϵ = 0. Based on this behaviour, we propose an interesting double-scaling limit which fixes λ ≡ Jϵ, suitable for probing the region of the crossover. I will give ∆(J; D) to next-to-leading order in perturbation theory, either in 1/λ or in λ, valid when λ ≫ 1 and λ ≪ 1, respectively.