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arXiv:1207.3670 · Physical Review E

Efimov effect of triple-stranded DNA: Real-space renormalization group and zeros of the partition function

Jaya Maji, Somendra M. Bhattacharjee

Published Oct 25, 2012• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.86.041147

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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.
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.
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.
We study the semisuper-Efimov effect, which is found for four identical bosons with a resonant three-body interaction in 2D, in various systems. Based on solutions of bound-state and renormalization-group equations, we first demonstrate an emergence of the semisuper-Efimov effect in mass-imbalanced bosons in 2D. Compared with the Efimov and the super-Efimov effects, the mass ratio-dependent scaling parameter is unexpectedly found to take on a finite value even for extremely mass-imbalanced situations, where the mass ratio is 0 or ∞\infty∞. By a renormalization-group analysis, we also show that a weak two-body interaction sustains the semisuper-Efimov effect. Finally, we liberate the universality of the semisuper-Efimov effect from 2D by showing that bosons with linear-dispersion relation support the semisuper-Efimov effect in 1D.
In the low-energy limit, non-relativistic particles with short-range interactions exhibit universal behavior that is largely independent of microscopic details. This universality is typically described by effective field theory, in which the two-body interaction is renormalized to a single parameter-the scattering length. For systems of identical bosons, the three-body problem reveals the Efimov effect, a novel phenomenon proposed that necessitates the introduction of an additional three-body parameter. However, the exact relation between this three-body parameter, the coupling constants in the effective field theory, and the binding energies of Efimov states remains unresolved. In this Letter, we address this question through a comprehensive analysis of the Skorniakov-Ter-Martirosian equation with a finite cutoff. We establish an exact renormalization relation for the three-body parameter and determine its connection to the energies of Efimov bound states. These results are validated through high-precision numerical simulations. We expect our findings to be of fundamental interest across various fields, including atomic, nuclear, condensed matter, and particle physics, and to have broad applications in both few-body and many-body physics.