Limit Cycles in Quantum Theories

Głazek, Stanisław D.; Wilson, Kenneth G.

doi:10.1103/physrevlett.89.230401

Cited by 91 publications

(69 citation statements)

References 14 publications

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“…We employ the functional renormalization-group (FRG) analysis, which allows a non-perturbative RG transformation onto the flowing action Γ k [ψ, ψ * , φ, φ * ], in which the coefficients Γ (3) k and Γ (4) k of |ψφ| 2 and |ψψφ| 2 are identified with the k-dependent particle-dimer and particle-particle-dimer correlation functions. The exact RG flow of Γ k [ψ, ψ * , φ, φ * ] is governed by the Wetterich equation [40] (equation (5) in Methods) and the flows of Γ (3) k and Γ (4) k can be extracted from the vertex expansion [41] (equation (7) in Methods) of the Wetterich equation.…”

Section: Topologically Connected 3-and 4-body Limit Cyclementioning

confidence: 99%

“…where Γ (n) k is the one-particle irreducible vertex that represents the correlation of n particles at the RG cutoff scale of k. By substituting equation (7) into equation (5), we obtain the exact FRG equation of Γ (n) k . Since we are interested in the 3-body and 4-body physics, we have only to consider the one-particle irreducible vertices up to n = 4.…”

Section: Vertex Expansionmentioning

confidence: 99%

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Topological origin of universal few-body clusters in Efimov physics

Horinouchi

Ueda

2016

Phys. Rev. A

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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]....

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Section: Topologically Connected 3-and 4-body Limit Cyclementioning

confidence: 99%

Section: Vertex Expansionmentioning

confidence: 99%

Topological origin of universal few-body clusters in Efimov physics

Horinouchi

Ueda

2016

Phys. Rev. A

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“…Głazek and Wilson [126,127] investigated the possibility of limit cycles. The connection between asymptotic freedom and limit cycles has been studied in [121,122].…”

Section: Flow Equations and Similarity Renormalizationmentioning

confidence: 99%

In Memory of Kenneth G. Wilson

Wegner

2014

J Stat Phys

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“…This is as it should be: the monotonicity of the flow of a implies that limit cycles do not exist in any physically meaningful sense [41,42]; in fact, they may be removed by a field and coupling constant redefinition. However, it is well known that bona-fide renormalization group limit cycles exist in some non-relativistic theories [43][44][45]. The C- The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

“…,(44),(45) are not linearly independent in the scheme with 2a + c = 2ρ 23 + c = χ 4 − c = χ 4 − p 4 = p 4 + ρ 23 = 0 and a 4α = ρ 1α = ρ 7α = χ 1α = 0.…”

mentioning

confidence: 99%

Weyl consistency conditions in non-relativistic quantum field theory

Pal

Grinstein

2016

J. High Energ. Phys.

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Weyl consistency conditions have been used in unitary relativistic quantum field theory to impose constraints on the renormalization group flow of certain quantities. We classify the Weyl anomalies and their renormalization scheme ambiguities for generic non-relativistic theories in 2 + 1 dimensions with anisotropic scaling exponent z = 2; the extension to other values of z are discussed as well. We give the consistency conditions among these anomalies. As an application we find several candidates for a C-theorem. We comment on possible candidates for a C-theorem in higher dimensions.

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Limit Cycles in Quantum Theories

Cited by 91 publications

References 14 publications

Topological origin of universal few-body clusters in Efimov physics

Topological origin of universal few-body clusters in Efimov physics

In Memory of Kenneth G. Wilson

Weyl consistency conditions in non-relativistic quantum field theory

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