Abstract
We study the transition between phases at large R-charge on a conformal manifold. These phases are characterized by the behaviour of the lowest operator dimension ∆(QR) for fixed and large R-charge QR. We focus, as an example, on the D = 3, mathcal{N} = 2 Wess-Zumino model with cubic superpotential W= XYZ+frac{tau }{6}left({X}^3+{Y}^3+{Z}^3right) , and compute ∆(QR, τ) using the ϵ-expansion in three interesting limits. In two of these limits the (leading order) result turns out to beΔQR,τ=BPSbound1+Oϵτ2QR,QR≪1ϵ1ϵτ298ϵτ22+τ21D−1QRDD−11+Oϵτ2QR−2D−1,QR≫1ϵ,1ϵτ2\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\Delta \\left({Q}_{R,\\tau}\\right)=\\left\\{\\begin{array}{ll}\\left(\\mathrm{BPS}\\;\\mathrm{bound}\\right)\\left[1+O\\left(\\epsilon {\\left|\\tau \\right|}^2{Q}_R\\right)\\right],& {Q}_R\\ll \\left\\{\\frac{1}{\\epsilon },\\kern0.5em \\frac{1}{\\epsilon {\\left|\\tau \\right|}^2}\\right\\}\\\\ {}\\frac{9}{8}{\\left(\\frac{\\epsilon {\\left|\\tau \\right|}^2}{2+{\\left|\\tau \\right|}^2}\\right)}^{\\frac{1}{D-1}}{Q}_R^{\\frac{D}{D-1}}\\left[1+O\\left({\\left(\\epsilon {\\left|\\tau \\right|}^2{Q}_R\\right)}^{-\\frac{2}{D-1}}\\right)\\right],& {Q}_R\\gg \\left\\{\\begin{array}{ll}\\frac{1}{\\epsilon },& \\frac{1}{\\epsilon {\\left|\\tau \\right|}^2}\\end{array}\\right\\}\\end{array}\\right. $$\\end{document}which leads us to the double-scaling parameter, ϵ|τ|2QR, which interpolates between the “near-BPS phase” (∆(Q) ∼ Q) and the “superfluid phase” (∆(Q) ∼ QD/(D−1)) at large R-charge. This smooth transition, happening near τ = 0, is a large-R-charge manifestation of the existence of a moduli space and an infinite chiral ring at τ = 0. We also argue that this behavior can be extended to three dimensions with minimal modifications, and so we conclude that ∆(QR, τ) experiences a smooth transition around QR ∼ 1/|τ|2. Additionally, we find a first-order phase transition for ∆(QR, τ) as a function of τ, as a consequence of the duality of the model. We also comment on the applicability of our result down to small R-charge.
Highlights
The large global charge sector of generic conformal field theories is known to simplify [1,2,3]
Quantum fluctuations of the theory become negligible. This is because adding such a chemical potential leads to a natural separation of scales, i.e. the ratio between the UV and the IR scales are given by ΛIR/ΛUV ∝ Q−α for some α > 0, where Q denotes the global charge which we take large
Note that the first regime would have been inaccessible by using the supersymmetry algebra as the operator we are interested in is usually way above the BPS bound
Summary
We will discuss the large-Q regime, were B = μ 1. We calculate the sum for small Q, where y = μ − m √ 1. We will compute it for intermediate Q where y = μ − m and y |τ |. 1. The sum simplifies in these limits, and they are precisely the limits which concern us. Note that the sums over the first D + 1 terms c1, . CD+1 diverge, and so they must be regulated. In the -expansion, we can expand each ck: ck = ck,0 + ck,1 + O( 2)
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