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arXiv:1409.8127 · Physical Review E

Efimov-like phase of a three-stranded DNA and the renormalization-group limit cycle

Tanmoy Pal, Poulomi Sadhukhan, Somendra M. Bhattacharjee

Published Apr 7, 2015• 10 citations
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Abstract

A three-stranded DNA with short range base pairings only is known to exhibit a classical analog of the quantum Efimov effect, viz., a three-chain bound state at the two-chain melting point where no two are bound. By using a nonperturbative renormalization-group method for a rigid duplex DNA and a flexible third strand, with base pairings and strand exchange, we show that the Efimov-DNA is associated with a limit cycle type behavior of the flow of an effective three-chain interaction. The analysis also shows that thermally generated bubbles play an essential role in producing the effect. A toy model for the flow equations shows the limit cycle in an extended three-dimensional parameter space of the two-chain coupling and a complex three-chain interaction.

DOI
10.1103/physreve.91.042105

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Theoretical study on a renormalization-group limit cycle in Efimov physics

裕理 堀之内

Physics is often said to be universal when microscopically distinct systems show the same lowenergy behavior.In critical phenomena, the universality of the second-order phase transition is elucidated by the renormalization-group (RG) method, which provides a powerful computational tool for the critical exponents of the power-law behavior of observables.In particular, the attraction of different RG-flow trajectories to a small-dimensional sub-theory space accounts for the universal nature.The attraction structure motivates us to extract universal physical observables by investigating the sub-theory space called the renormalized trajectory, onto which RG flows are attracted.To put the motivation forward, we consider the universality in quantum few-body physics, in which the RG exhibits another characteristic flow than the fixed points.The Efimov physics, which is renowned for self-similar trimer states, shows an RG limit-cycle behavior that refers to a periodic RG-flow trajectory.In this thesis, we investigate the relation between the RG limit cycle and the universality in Efimov physics.Firstly, to verify that the RG limit cycle is the renormalized trajectory in Efimov physics, we demonstrate that the RG-flow trajectories of microscopically distinct systems universally exhibit the RG limit-cycle behavior at sufficiently low energy, thereby giving a numerical evidence of the attraction structure of the RG-flow trajectories to the RG limit cycle; in other words, we show that the RG limit cycle behaves as an attractor of various RG flows.To address this point unbiasedly, we perform an exact RG calculation for various effective models of identical bosons that features Efimov physics at low energy.We apply an exact functional renormalization-group (FRG) method to effective separable models for various systems of identical bosons with short-range resonant interactions.For a given microscopic three-body model, we numerically demonstrate for the first time that the RG-flow trajectory converges at the RG limit cycle in the low-energy regime.Physical observables such as Efimov's scaling factor and the three-body parameter can be extracted from the RG limit cycle.Secondly, we investigate the role of RG limit cycle in a four-body extension of Efimov physics, which has attracted considerable attention in recent years.Although the universality in four-body physics has been established both theoretically and experimentally, the relationship between the RG limit cycle and the four-body physics remains an open question.By performing an FRG calculation for a simple effective field theory, we obtain for the first time the RG limit cycle in the four-body sector that reproduces the universality of the four-body physics.As the first attempt, we develop a simple separable pole approximation of the four-body FRG equation and successfully reproduce qualitative features of the four-body physics.A systematic improvement of the approximation quantitatively reproduces the numbers and the binding energies of tetramer states that emerge universally at low energy.2 The restriction r 0 < ∼ R < ∼ a on the length scale limits the binding energy E t of the Efimov trimers as ℏ ma 2 < ∼ E t < ∼ ℏ mr 2 0 .3 A similar discrete scale-invariant phenomenon is known for the doubly excited H -ion, in which two valence electrons are excited from their ground states [16].In the doubly excited H -ion, an interaction between an electron-

•Mar 23, 2017•0 citations
A three-stranded DNA with short range base pairings only is known to exhibit a classical analog of the quantum Efimov effect, viz., a three-chain bound state at the two-chain melting point where no two are bound. By using a nonperturbative renormalization-group method for a rigid duplex DNA and a flexible third strand, with base pairings and strand exchange, we show that the Efimov-DNA is associated with a limit cycle type behavior of the flow of an effective three-chain interaction. The analysis also shows that thermally generated bubbles play an essential role in producing the effect. A toy model for the flow equations shows the limit cycle in an extended three-dimensional parameter space of the two-chain coupling and a complex three-chain interaction.
We define a generalized model for three-stranded DNA consisting of two chains of one type and a third chain of a different type. The DNA strands are modeled by random walks on the three-dimensional cubic lattice with different interactions between two chains of the same type and two chains of different types. This model may be thought of as a classical analog of the quantum three-body problem. In the quantum situation, it is known that three identical quantum particles will form a triplet with an infinite tower of bound states at the point where any pair of particles would have zero binding energy. The phase diagram is mapped out, and the different phase transitions are examined using finite-size scaling. We look particularly at the scaling of the DNA model at the equivalent Efimov point for chains up to 10 000 steps in length. We find clear evidence of several bound states in the finite-size scaling. We compare these states with the expected Efimov behavior.
We study the melting of three-stranded DNA by using the real-space renormalization group and exact recursion relations. The prediction of an unusual Efimov-analog three-chain bound state, that appears at the critical melting of two-chain DNA, is corroborated by the zeros of the partition function. The distribution of the zeros has been studied in detail for various situations. We show that the Efimov DNA can occur even if the three-chain (i.e., three-monomer) interaction is repulsive in nature. In higher dimensions, a striking result that emerged in this repulsive zone is a continuous transition from the critical state to the Efimov DNA.
The aim of this thesis is to investigate, using simplified lattice models, whether the Efimov effect, well known in quantum mechanics, can arise in polymer physics too, particularly in a triple stranded DNA system. Such effect consists in the formation of stable trimer bound states when dimer bound states are not stable. We base strand interaction rules on a Poland-Scheraga model for a directed DNA-like polymer, double-stranded first and triple-stranded later. Within the Poland-Scheraga scheme, only interactions between base pairs with the same monomer indexes along the strands are permitted. Thus, the identification of the monomer index with imaginary time allows the formal mapping to the quantum problem in which particles interact at the same time. According to the transfer matrix method, we are able to extract information about the free-energy of the system from matrix eigenvalues. Studying them at the critical point of the unzipped-zipped phase transition we look for the similarities between the bound states in the polymer model and those predicted by Efimov theory for a three particle quantum system. In particular, in Efimov theory, an infinite series of trimer energy levels, with a constant ratio between consecutive levels, is found at the critical point of the 2-body problem.
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.
Physics is often said to be universal when microscopically distinct systems show the same lowenergy behavior.In critical phenomena, the universality of the second-order phase transition is elucidated by the renormalization-group (RG) method, which provides a powerful computational tool for the critical exponents of the power-law behavior of observables.In particular, the attraction of different RG-flow trajectories to a small-dimensional sub-theory space accounts for the universal nature.The attraction structure motivates us to extract universal physical observables by investigating the sub-theory space called the renormalized trajectory, onto which RG flows are attracted.To put the motivation forward, we consider the universality in quantum few-body physics, in which the RG exhibits another characteristic flow than the fixed points.The Efimov physics, which is renowned for self-similar trimer states, shows an RG limit-cycle behavior that refers to a periodic RG-flow trajectory.In this thesis, we investigate the relation between the RG limit cycle and the universality in Efimov physics.Firstly, to verify that the RG limit cycle is the renormalized trajectory in Efimov physics, we demonstrate that the RG-flow trajectories of microscopically distinct systems universally exhibit the RG limit-cycle behavior at sufficiently low energy, thereby giving a numerical evidence of the attraction structure of the RG-flow trajectories to the RG limit cycle; in other words, we show that the RG limit cycle behaves as an attractor of various RG flows.To address this point unbiasedly, we perform an exact RG calculation for various effective models of identical bosons that features Efimov physics at low energy.We apply an exact functional renormalization-group (FRG) method to effective separable models for various systems of identical bosons with short-range resonant interactions.For a given microscopic three-body model, we numerically demonstrate for the first time that the RG-flow trajectory converges at the RG limit cycle in the low-energy regime.Physical observables such as Efimov's scaling factor and the three-body parameter can be extracted from the RG limit cycle.Secondly, we investigate the role of RG limit cycle in a four-body extension of Efimov physics, which has attracted considerable attention in recent years.Although the universality in four-body physics has been established both theoretically and experimentally, the relationship between the RG limit cycle and the four-body physics remains an open question.By performing an FRG calculation for a simple effective field theory, we obtain for the first time the RG limit cycle in the four-body sector that reproduces the universality of the four-body physics.As the first attempt, we develop a simple separable pole approximation of the four-body FRG equation and successfully reproduce qualitative features of the four-body physics.A systematic improvement of the approximation quantitatively reproduces the numbers and the binding energies of tetramer states that emerge universally at low energy.2 The restriction r 0 < ∼ R < ∼ a on the length scale limits the binding energy E t of the Efimov trimers as ℏ ma 2 < ∼ E t < ∼ ℏ mr 2 0 .3 A similar discrete scale-invariant phenomenon is known for the doubly excited H -ion, in which two valence electrons are excited from their ground states [16].In the doubly excited H -ion, an interaction between an electron-