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Weyl consistency conditions in non-relativistic quantum field theory

Sridip Pal, Grinstein, Benjamín

Published Dec 1, 2016• 16 citations
Publisher

Abstract

Weyl consistency conditions have been used in unitary relativistic quantum\nfield theory to impose constraints on the renormalization group flow of certain\nquantities. We classify the Weyl anomalies and their renormalization scheme\nambiguities for generic non-relativistic theories in 2+1 dimensions with\nanisotropic scaling exponent z=2; the extension to other values of z are\ndiscussed as well. We give the consistency conditions among these anomalies. As\nan application we find several candidates for a C-theorem. We comment on\npossible candidates for a $C$-theorem in higher dimensions.

DOI
10.1007/jhep12%282016%29012

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Weyl consistency conditions have been used in unitary relativistic quantum\nfield theory to impose constraints on the renormalization group flow of certain\nquantities. We classify the Weyl anomalies and their renormalization scheme\nambiguities for generic non-relativistic theories in 2+1 dimensions with\nanisotropic scaling exponent z=2; the extension to other values of z are\ndiscussed as well. We give the consistency conditions among these anomalies. As\nan application we find several candidates for a C-theorem. We comment on\npossible candidates for a CCC-theorem in higher dimensions.
Weyl consistency conditions have been used in unitary relativistic quantum field theory to impose constraints on the renormalization group flow of certain quantities. We classify the Weyl anomalies and their renormalization scheme ambiguities for generic non-relativistic theories in 2 + 1 dimensions with anisotropic scaling exponent z = 2; the extension to other values of z are discussed as well. We give the consistency conditions among these anomalies. As an application we find several candidates for a C-theorem. We comment on possible candidates for a C-theorem in higher dimensions.
The concept of renormalization group (RG) flow is one of the most novel and broad-reaching aspects of quantum field theory (QFT). The RG flow is implemented by constructing effective descriptions of a QFT at decreasing energy scales. One reason that RG flow is useful is that often one is interested in low-energy properties of theories with complicated short-distance structures. RG flows are subject to C theorems in relativistic QFT. The C theorems order the space of Lorentz-invariant QFTs. RG flows generically begin at scale-invariant fixed points known as conformal field theories (CFTs) and end in trivial massive theories. With tuning, the RG flows may end at non-trivial CFTs. Each CFT has an associated dimensionless C value. The C theorem states that under RG flow from a UV to an IR fixed point, the C value decreases. In this Dissertation I present the F-theorem, which is a C theorem in three spacetime dimensions. I show that the correct quantity to consider is the Euclidean free energy of the CFT conformally mapped to the three-sphere, known as the F value. After motivating the F-theorem, I develop tools for calculating the F value in a variety of CFTs, with and without supersymmetry, including free field theories and gauge theories with large numbers of flavors. I also show that the F value is itself a useful quantity for probing the gauge/gravity duality and understanding other aspects of CFT, such as the scaling dimensions of monopole operators. The F theorem is closely related to quantum entanglement entropy. At conformal fixed points, the F value is equal to minus the renormalized entanglement entropy (REE) in flat Minkowski space across a circle. Away from the fixed points, the REE is a monotonically decreasing function along the RG flow. I compute the REE in a variety of holographic and non-holographic theories. I conclude the Dissertation by discussing a somewhat surprising result: the REE is not stationary at conformal fixed points.
Abstract We study conformal field theories with boundaries, and their boundary renormalization group (RG) flows, using methods from quantum information theory. Positivity of the relative entropy, together with unitarity and Lorentz invariance, give rise to bounds that characterize the irreversibility of such flows. This generalizes the recently proved entropic g-theorem to higher dimensions. In 2 + 1 dimensions with a boundary, we prove the entropic b-theorem — the decrease of the two-dimensional Weyl anomaly under boundary RG flows. In higher dimensions, the bound implies that the leading area coefficient of the entanglement entropy induced by the defect decreases along the flow. Our proof unifies these properties, and provides an information-theoretic interpretation in terms of the distinguishability between the short distance and long distance states. Finally, we establish a sum rule for the change in the area term in theories with boundaries, which could have implications for models with localized gravity.
A bstract We study applications of spectral positivity and the averaged null energy condition (ANEC) to renormalization group (RG) flows in two-dimensional quantum field theory. We find a succinct new proof of the Zamolodchikov c -theorem, and derive further independent constraints along the flow. In particular, we identify a natural C -function that is a completely monotonic function of scale, meaning its derivatives satisfy the alternating inequalities (–1) n C ( n ) ( μ 2 ) ≥ 0. The completely monotonic C -function is identical to the Zamolodchikov C -function at the endpoints, but differs along the RG flow. In addition, we apply Lorentzian techniques that we developed recently to study anomalies and RG flows in four dimensions, and show that the Zamolodchikov c -theorem can be restated as a Lorentzian sum rule relating the change in the central charge to the average null energy. This establishes that the ANEC implies the c -theorem in two dimensions, and provides a second, simpler example of the Lorentzian sum rule.
We employ the recent results on the generalization of the central charge theorem to three spacetime dimensions to derive nonperturbative results for several strongly interacting quantum field theories, including quantum electrodynamics (QED-3), and the theory corresponding to certain quantum phase transitions in condensed matter systems. In particular, by demanding that the universal constant part of the entanglement entropy decreases along the renormalization group flow (F theorem), we find sufficient conditions for the stability of QED-3 against chiral symmetry breaking and confinement. Using similar ideas, we derive strong constraints on the nature of quantum critical points in condensed matter systems with topological order.