A note on inhomogeneous ground states at large global charge

159

We study the scaling dimension ∆ φ n of the operator φ n where φ is the fundamental complex field of the U (1) model at the Wilson-Fisher fixed point in d = 4 − ε. Even for a perturbatively small fixed point coupling λ * , standard perturbation theory breaks down for sufficiently large λ * n. Treating λ * n as fixed for small λ * we show that ∆ φ n can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in We explicitly compute the first two orders in the expansion, ∆ −1 (λ * n) and ∆ 0 (λ * n). The result, when expanded at small λ * n, perfectly agrees with all available diagrammatic computations. The asymptotic at large λ * n reproduces instead the systematic large charge expansion, recently derived in CFT. Comparison with Monte Carlo simulations in d = 3 is compatible with the obvious limitations of taking ε = 1, but encouraging.

Section: Semiclassics At Small Fixed λNmentioning

confidence: 73%

Section: Semiclassics At Small Fixed λNmentioning

confidence: 78%

Section: · · ·mentioning

confidence: 99%

Section: One-loop Correction: ∆mentioning

confidence: 99%

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The epsilon expansion meets semiclassics

Badel

Cuomo

Monin

et al. 2019

159

“…Now a natural question is to ask whether the result is valid when ∆ and J is not of the same order. For example, in the large charge expansion literature [9][10][11][12][13][14][15][16][17][18], the regime where J ∆ 1/n with n > 1 is being probed. It turns out that only a part of the answer coming from the naive Cardy like analysis is meaningful while the rest of it is comparable to the error term.…”

Section: Contentsmentioning

confidence: 99%

Tauberian-Cardy formula with spin

Pal

Sun

2020

We prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory. The asymptotic density of states with conformal weight (h,h) → (∞, ∞) for any arbitrary spin is derived using the theorem. We further rigorously show that the error term is controlled by the twist parameter and insensitive to spin. The sensitivity of the leading piece towards spin is discussed. We identify a universal piece in microcanonical entropy when the averaging window is large. An asymptotic spectral gap on (h,h) plane, hence the asymptotic twist gap is derived. We prove an universal inequality stating that in a compact unitary 2D CFT without any conserved current Ag ≤ π(c−1)r 2 24 is satisfied, where g is the twist gap over vacuum and A is the minimal "areal gap", generalizing the minimal gap in dimension to (h ,h ) plane and r = 4 √ 3 π 2.21. We investigate density of states in the regime where spin is parametrically larger than twist with both going to infinity. Moreover, the large central charge regime is studied. We also probe finite twist, large spin behavior of density of states.

Transition of large R-charge operators on a conformal manifold

Sharon

Watanabe

2021

Self Cite

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, $$ \mathcal{N} $$ N = 2 Wess-Zumino model with cubic superpotential $$ W= XYZ+\frac{\tau }{6}\left({X}^3+{Y}^3+{Z}^3\right) $$ W = XYZ + τ 6 X 3 + Y 3 + Z 3 , and compute ∆(QR, τ) using the ϵ-expansion in three interesting limits. In two of these limits the (leading order) result turns out to be$$ \Delta \left({Q}_{R,\tau}\right)=\left\{\begin{array}{ll}\left(\mathrm{BPS}\;\mathrm{bound}\right)\left[1+O\left(\epsilon {\left|\tau \right|}^2{Q}_R\right)\right],& {Q}_R\ll \left\{\frac{1}{\epsilon },\kern0.5em \frac{1}{\epsilon {\left|\tau \right|}^2}\right\}\\ {}\frac{9}{8}{\left(\frac{\epsilon {\left|\tau \right|}^2}{2+{\left|\tau \right|}^2}\right)}^{\frac{1}{D-1}}{Q}_R^{\frac{D}{D-1}}\left[1+O\left({\left(\epsilon {\left|\tau \right|}^2{Q}_R\right)}^{-\frac{2}{D-1}}\right)\right],& {Q}_R\gg \left\{\begin{array}{ll}\frac{1}{\epsilon },& \frac{1}{\epsilon {\left|\tau \right|}^2}\end{array}\right\}\end{array}\right. $$ Δ Q R , τ = BPS bound 1 + O ϵ τ 2 Q R , Q R ≪ 1 ϵ 1 ϵ τ 2 9 8 ϵ τ 2 2 + τ 2 1 D − 1 Q R D D − 1 1 + O ϵ τ 2 Q R − 2 D − 1 , Q R ≫ 1 ϵ , 1 ϵ τ 2 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.