Scale anomaly of a Lifshitz scalar: A universal quantum phase transition to discrete scale invariance

Abstract

We demonstrate the existence of a universal transition from a continuous scale invariant phase to a discrete scale invariant phase for a class of one-dimensional quantum systems with anisotropic scaling symmetry between space and time. These systems describe a Lifshitz scalar interacting with a background potential. The transition occurs at a critical coupling λc corresponding to a strongly attractive potential.

A New Version of the Aharonov–Bohm Effect

“…Such choice corresponds, mathematically, to the choice of a self-adjoint extension of the initial magnetic Hamiltonian, since usually the preliminary physical information provides only a symmetric operator; the self-adjoint property is required (in fact equivalent) for a unitary time evolution (conservation of probability) in quantum dynamics. The topic is related to the existence of anomalies [13], the presence of anisotropic scale invariances [14], different surface spectra of Weyl semimetals [35], creation of a pointlike source in quantum field theory [39], studies of topological quantum phases [3] and models in quantum gravity whose time evolution depend on boundary conditions at the origin [28], to mention only a handful of examples that illustrate the well-known fact that different self-adjoint extensions correspond to different physics. An instructive discussion about self-adjoint extensions of the simple case of a particle in a potential well appears in [12].…”

Complex dimensions and their observability

“…Note added.-After submission, [53] appeared where the authors discussed a phase transition between a continuous and a discrete scale invariance in Lifshitz-type Hamiltonian models. After some elaboration, these results can have a direct application to Hořava-Lifshitz gravity and corroborate the main claim of this paper, providing another and important explicit example of how discrete scale invariance (and complex dimensions) can emerge naturally in quantum gravities.…”

Discrete Scale Invariance in Holography Revisited

“…This type of symmetry, respectively the associated signatures, can be exhibited by a variety of systems such as fractals (for example the famous Koch curve or the triadic Cantor set), stock markets, earthquakes, black hole formation, perturbations of extremal black holes (see e.g. []), black hole/black string phase transitions, the Efimov effect, QFT toy models, quantum gravity, condensed matter models and even holographic AdS/CMT models . See [] for a general review.…”

“…In order to learn more about cyclic RG flows and discrete scale invariance, it might be useful to construct holographic models which exhibit this type of symmetry. As already mentioned above, there are a few holographic models in which log‐periodic oscillations of certain variables are known to occur. In the models, however, these log‐periodic oscillations did not occur on a spacetime axis, but instead as a function of a frequency or temperature.…”