Topological origin of universal few-body clusters in Efimov physics

Abstract

Efimov physics is renowned for the self-similar spectrum featuring the universal ratio of one eigenenergy to its neighbor. Even more esoteric is the numerically unveiled fact that every Efimov trimer is accompanied by a pair of tetramers. Here we demonstrate that this hierarchy of universal few-body clusters has a topological origin by identifying the numbers of universal 3-and 4-body bound states with the winding numbers of the renormalization-group limit cycle in theory space. The finding suggests a topological phase transition in mass-imbalanced few-body systems which should be tested experimentally.Universality in physics often refers to a situation in which apparently distinct systems show the same lowenergy behavior. A prominent example is the critical phenomena, where different physical systems are grouped into a set of universality classes sharing the same critical exponents. The modern foundation for understanding the universality is established by the renormalization group (RG) [1], which allows us to investigate a change of a system viewed at different distance scales by following an RG flow of system-parameters generated by a recursive coarse graining of the system. In particular, the universality classes of critical phenomena can be categorized by the fixed points of the RG flow, which represent the scale invariance of the second-order phase transition at which no characteristic length scale is present.Yet another characteristic RG flow can be found in universal quantum few-body physics that feature discrete scale invariance. Here we consider the Efimov effect [2] and its 4-body extension [3][4][5] in which resonantly interacting 3 and 4 bosons form an infinite series of universal 3-and 4-body bound states that feature the self-similar spectrum: the 3-body bound states (Efimov trimers) are related to one another by a scaling factor of (22.7) 2 , and each Efimov trimer is accompanied by two 4-body bound states (see Fig. 1). This discrete scale invariance makes the Efimov physics a prime example of the RG limit cycle [6][7][8][9][10], where an RG flow forms a periodic circle rather than converges to a fixed point. Because of the universality and the uniqueness, much theoretical [2, 11-16] and experimental [17][18][19][20][21][22][23][24][25][26][27] efforts have been devoted to reveal the existence of the Efimov trimers in a rich variety of systems. Also, the existence of the 4-body companions associated with Efimov trimers has been confirmed both numerically [3][4][5][28][29][30][31] and experimentally [32].Despite such extensive research, there remains an as yet unresolved fundamental question: How is the universal 4-body physics related to the RG limit cycle? In the following, we answer this question by showing that * E-mail: yusuke@cat.phys.s.u-tokyo.ac.jp 4b /E 3b ≃ 4.58, respectively [5]. A more detailed energy spectrum can be found in Refs. [29][30][31].the hierarchical structure of the few-body clusters has a topological origin in terms of the RG limit cycle, as we conjectured previously [33]....

Window for Efimov physics for few-body systems with finite-range interactions

“…This has been confirmed experimentally [6,14], which therefore raises the question, whether this pattern continues for higher N . Theoretical calculations for N = 5, 6 have been made, which predict that the pattern of two N + 1 state accompany each universal N ground state at least for N ≤ 6 [15][16][17][18][19][20][21][22][23][24]. Experimental evidence that two Borromean pentamer (N = 5) states does accompany the ground tetramer state have been reported [25].…”

Unitarity and Discrete Scale Invariance

“…with similar implications as for N = 3. I am not aware of an explanation for the doubling, which has a topological interpretation [72], but the replicating towers are a reflection of the surviving discrete scale invariance. For N ≥ 4, all but the two lower states appear as resonances in the scattering of a particle on the (N − 1)-particle ground state.…”

Naturalness in nuclear effective field theories

“…For a given A, there are 2 (A−3) interlocking towers of states. The replicating towers are a reflection of DSI, but the doubling, which has a topological interpretation [89], is of unclear origin. For A ≥ 5 four-component fermions the pattern of towers is not known.…”