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 $\lambda_{c}$ corresponding to a strongly attractive potential.
Highlights
Some of the most intriguing phenomena resulting from quantum physics are the violation of classical symmetries, collectively referred to as anomalies [1,2,3,4]
One class of anomalies describes the breaking of continuous scale symmetry at the quantum level
The existence and geometric structure of such levels do not rely on the details of the potential close to its source and is a signature of residual pdisffifficffiffirffiffieffiffitffieffiffiffi scale invariance since fEng → fexp ð−2π=
Summary
Some of the most intriguing phenomena resulting from quantum physics are the violation of classical symmetries, collectively referred to as anomalies [1,2,3,4]. Scale invariant [24], any system described by the Hamiltonian, H S 1⁄4 p2=2m − λ=r2, exhibits an abrupt transition in the spectrum at 2mλc 1⁄4 ðd − 2Þ2=4 [25] where d is the space dimension. The similarity between the spectra and transition of these Dirac and Schrödinger Hamiltonians motivates the study of whether a transition of this sort is possible for a generic scale invariant system. Hamiltonian (1) describes a system with non-quadratic anisotropic scaling between space and time for N > 1 This “Lifshitz scaling symmetry” [32], manifest in (1), can be seen for example at the finite temperature multicritical points of certain materials [33,34] or in strongly correlated electron systems [35,36,37]. We analyze the x0 1⁄4 0 case, obtain its self adjoint extensions and spectrum and obtain similar results
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