Large charge at large N
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Springer
Received: September 26, Revised: November 24, Accepted: December 6, Published: December 19,
2019 2019 2019 2019
Luis Alvarez-Gaume,a,b Domenico Orlandoc,d and Susanne Reffertd a
Simons Center for Geometry and Physics, State University of New York, Stony Brook, NY–11794–3636, U.S.A. b Theory Department, CERN, ch-1211 Geneva 23, Switzerland c INFN sezione di Torino, Arnold-Regge Center, via Pietro Giuria 1, 10125 Turin, Italy d Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
E-mail: [email protected], [email protected], [email protected] Abstract: We apply the large-charge expansion to O(N ) vector models starting from first principles, focusing on the Wilson-Fisher point in three dimensions. We compute conformal dimensions at zero and finite temperature at fixed charge Q, concentrating on the regime 1 N Q. Our approach places the earlier effective field theory treatment on firm ground and extends its predictions. Keywords: 1/N Expansion, Conformal Field Theory, Global Symmetries, Spontaneous Symmetry Breaking ArXiv ePrint: 1909.02571
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2019)142
JHEP12(2019)142
Large charge at large N
Contents 1 Introduction
1
2 The action 2.1 The infrared fixed point 2.2 The effective action for λ
4 4 6
saddle point Saddle point equations Zeta-function regularization Zero charge, zero temperature: the conformal coupling Finite charge, zero temperature: the broken phase Finite charge, finite temperature: the unbroken phase
8 8 9 10 12 16
4 The Goldstones
19
5 Conclusions
23
A The zeta function on Sβ1 × Σ
24
B The zeta function on the torus and the two-sphere
26
ˆ C The one-loop term in the propagator for λ
29
1
Introduction
The study of quantum systems in the limit of large quantum numbers goes back to the early years of quantum theory in terms of the wkb approximation. In theories without an obvious expansion parameter, in many conformal field theories (CFTS) and in strongly coupled systems, it is often useful to look for special variables defining them, which under certain circumstances allow a new approximation scheme for the physical problems of interest. This is for example the case in the original semiclassical expansion in Quantum Mechanics, the large-N limit in quantum field theory (QFT) [1]1 and large-spin limits [3, 4]. Recently, there has been a lot of interest in the study of QFT with global symmetries in the limit of large charge. For several CFT, the anomalous dimensions of primary operators with large charge have been computed [5–9]. 1
Yaffe [2] discusses the sense in which large-N limits of various quantum theories are equivalent to classical limits.
–1–
JHEP12(2019)142
3 The 3.1 3.2 3.3 3.4 3.5
The coefficients c3/2 and c1/2 depend on the theory, but the constant term c0 ≈ −0.0937254 is universal and is a prediction of the theory.2 In [6] the analysis in [5] was
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