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arXiv:2402.10792 · Few-Body Systems

Discrete Scaling in Non-integer Dimensions

T. Frederico, R.M. Francisco, D. S. Rosa, Gastão I. Krein, Marcelo Takeshi Yamashita

Published Mar 19, 2024• 0 citations
arXivHTMLPDF
DOI
10.1007/s00601-024-01895-4

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We have recently shown that the critical Anderson electron in D=3 dimensions effectively occupies a spatial region of the infrared (IR) scaling dimension dIR≈8/3. Here, we inquire about the dimensional substructure involved. We partition space into regions of equal quantum occurrence probabilities, such that the points comprising a region are of similar relevance, and calculate the IR scaling dimension d of each. This allows us to infer the probability density p(d) for dimension d to be accessed by the electron. We find that p(d) has a strong peak at d very close to two. In fact, our data suggest that p(d) is non-zero on the interval [dmin,dmax]≈[4/3,8/3] and may develop a discrete part (δ-function) at d=2 in the infinite-volume limit. The latter invokes the possibility that a combination of quantum mechanics and pure disorder can lead to the emergence of integer (topological) dimensions. Although dIR is based on effective counting, of which p(d) has no a priori knowledge, dIR≥dmax is an exact feature of the ensuing formalism. A possible connection of our results to the recent findings of dIR≈2 in Dirac near-zero modes of thermal quantum chromodynamics is emphasized.
• Induced Efimov effect. • Confined three-body systems. • Non-integer dimensions. The Efimov effect can be induced by means of an external deformed one-body field that effectively reduces the allowed spatial dimensions to less than three. To understand this new mechanism, conceptually and practically, we employ a formulation using non-integer dimension, which is equivalent to the strength of an external oscillator field. The effect most clearly appears when the crucial two-body systems are unbound in three, but bound in two, dimensions. We discuss energy variation, conditions for occurrence, and number of Efimov states, as functions of the dimension. We use practical examples from cold atom physics of 133 Cs- 133 Cs- 133 Cs, 87 Rb- 87 Rb- 87 Rb, 133 Cs- 133 Cs- 6 Li, and 87 Rb- 87 Rb- 39 K. Laboratory tests of the effect can be performed with two independent parameters, i.e. the external one-body field and the Feshbach two-body tuning. The scaling and (dis)appearance of these Efimov states occur precisely as already found in three dimensions.
In this work, the quantum field theory that describes non-Fermi liquids realized at antiferromagnetic quantum critical points in space dimensions between two and three has been solved nonperturbatively. Using the solvable theory as a model system, the dimensional regularization scheme is put to the test. While the perturbative solution proves to be useful in constructing the nonperturbative solution, it reveals one subtlety: the low-energy limit and the integer dimension limits do not commute due to crossover energy scales that vanish at the physical dimensions.