The Efimov effect is the only experimentally realized universal phenomenon that exhibits the renormalization-group limit cycle with the three-body parameter parametrizing a family of universality classes. Recent experiments in ultracold atoms have unexpectedly revealed that the three-body parameter itself is universal when measured in units of an effective range. By performing an exact functional renormalization-group analysis with various finite-range interaction potentials, we demonstrate that the onset of the renormalization-group flow into the limit cycle is universal, regardless of short-range details, which connects the missing link between the two universalities of the Efimov physics. A close connection between the topological property of the limit cycle and few-body physics is also suggested.
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]....
Bulk properties of quantum phases should be independent of a specific choice of boundary conditions as long as the boundary respects the symmetries. Based on this physically reasonable requirement, we discuss the Lieb-Schultz-Mattis-type ingappability in two-dimensional quantum magnets under a boundary condition that makes evident a quantum anomaly underlying the lattice system. In particular, we direct our attention to those on the checkerboard lattice which are closely related to frustrated quantum magnets on the square lattice and on the Shastry-Sutherland lattice. Our discussion is focused on the adiabatic U(1) flux insertion through a closed path in a boundary condition twisted by a spatial rotation and a reflection. Two-dimensional systems in this boundary condition are effectively put on a nonorientable space, namely the Klein bottle. We show that the translation symmetry on the Klein-bottle space excludes the possibility of the unique and gapped ground state. Taking advantage of the flux insertion argument, we also discuss the ground-state degeneracy on magnetization plateaus of the Heisenberg antiferromagnet on the checkerboard lattice.
The sentence "Yao and Oshikawa reported quite recently a paper that follows this line [11]" cites a paper [11] by Y. Yao, C.-T. Hsieh, and M. Oshikawa. This sentence should cite another paper by Y. Yao and M. Oshikawa [1]. The same mistakes of citation are found in the original article as shown below. (ii) P. 2, the first paragraph: The sentence "In that context, we discuss the LSM-type ingappability on the checkerboard lattice as a continuation of the work in Refs. [11,12]" should cite Refs. [1,2]. (iii) P. 3, above Eq. (11): The sentence "Let us impose the tilted boundary condition on the system [11]" should cite Ref. [1] instead of [11] of the original article. (iv) P. 6, the first paragraph of Sec. VII: The citation of Ref. [11] in the sentence "Nevertheless, the well-known flux insertion argument turned out not to demonstrate the anomaly explicitly in the periodic [3] or the tilted [11] boundary conditions" should be replaced as Ref. [1]. (v) P. 7, the final paragraph: The sentence "Just as Ref. [11] did in the tilted boundary condition, we can extend • • • " should cite Ref. [1] instead of Ref. [11] of the original article.
We discuss the ground-state degeneracy of spin-1/2 kagome-lattice quantum antiferromagnets on magnetization plateaus by employing two complementary methods: the adiabatic flux insertion in closed boundary conditions and a 't Hooft anomaly argument on inherent symmetries in a quasi-onedimensional limit. The flux insertion with a tilted boundary condition restricts the lower bound of the ground-state degeneracy on 1/9, 1/3, 5/9, and 7/9 magnetization plateaus under the U(1) spinrotation and the translation symmetries: 3, 1, 3, and 3, respectively. This result motivates us further to develop an anomaly interpretation of the 1/3 plateau. Taking advantage of the insensitivity of anomalies to spatial anisotropies, we examine the existence of the unique gapped ground state on the 1/3 plateau from a quasi-one-dimensional viewpoint. In the quasi-one-dimensional limit, kagome antiferromagnets are reduced to weakly coupled three-leg spin tubes. Here, we point out the following anomaly description of the 1/3 plateau. While a simple S = 1/2 three-leg spin tube cannot have the unique gapped ground state on the 1/3 plateau because of an anomaly between a Z3 × Z3 symmetry and the translation symmetry at the 1/3 filling, the kagome antiferromagnet breaks explicitly one of the Z3 symmetries related to a Z3 cyclic transformation of spins in the unit cell. Hence the kagome antiferromagnet can have the unique gapped ground state on the 1/3 plateau.
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