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arXiv:1805.07973

A Study of Asymptotic Freedom like Behavior for Topological States of Matter

Ranjith Kumar R, S Rahul, Surya Narayan, Sujit Sarkar

Published May 21, 2018• 0 citations
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Abstract

We present results for asymptotic freedom like behavior for the topological state of the helical spin liquid system with finite proximity induced superconducting gap. We derive two different quantum Berezinskii-Kosterlitz-Thouless (BKT) equations for the two different limit of this model Hamiltonian. The common quantum phase for these two quantum BKT transitions is the helical Luttinger liquid phase where there is no evidence of asymptotic freedom. There is no evidence of superconductor-insulator transition for this asymptotic freedom study. We observe the evidance of asymptotic freedom for the two model Hamiltonian, but the character of the asymptotic phases are different. We also observe that the Luttinger liquid parameter plays a significant role to determine the asymptotic freedom but the chemical potential has no effect on it.

DOI
10.48550/arxiv.1805.07973

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We present results for asymptotic freedom like behavior for the topological state of the helical spin liquid system with finite proximity induced superconducting gap. We derive two different quantum Berezinskii-Kosterlitz-Thouless (BKT) equations for the two different limit of this model Hamiltonian. The common quantum phase for these two quantum BKT transitions is the helical Luttinger liquid phase where there is no evidence of asymptotic freedom. There is no evidence of superconductor-insulator transition for this asymptotic freedom study. We observe the evidance of asymptotic freedom for the two model Hamiltonian, but the character of the asymptotic phases are different. We also observe that the Luttinger liquid parameter plays a significant role to determine the asymptotic freedom but the chemical potential has no effect on it.
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The physics of the topological state of matter is the second revolution in quantum mechanics. We study the effect of interactions on the topological quantum phase transition and the quantum Berezinskii-Kosterlitz-Thouless (QBKT) transition in topological state of a quantum many-body condensed matter system. We predict a topological quantum phase transition from topological superconducting phase to an insulating phase for the interacting Kitaev chain. We observe interesting behaviour from the results of renormalization group study on the topological superconducting phase. We derive the renormalization group (RG) equation for QBKT through different routes with a few exact solutions along with the physical explanations, wherein we find the existence of two new important emergent phases apart from the two conventional phases of this model Hamiltonian. We also present results of a length-scale dependent study to predict asymptotic freedom like behaviour of the system. We do rigorous quantum field theoretical renormalization group calculations to solve this problem.
The classical Heisenberg model is one of the most fundamental models in statistical and condensed matter physics. Extensive theoretical and numerical studies suggest that, in two dimensions, this model does not exhibit a finite-temperature phase transition but instead manifests asymptotic freedom. However, some research has also proposed the possibility of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition over the years. In this study, we revisit the classical two-dimensional (2D) Heisenberg model through large-scale simulations with linear system sizes up to L=16384L=16384L=16384. Our Monte Carlo data, without any extrapolation, clearly reveal an exponential divergence of the correlation length $\ensuremath{\xi}$ as a function of the inverse temperature $\ensuremath{\beta}$, a hallmark of asymptotic freedom. Moreover, extrapolating $\ensuremath{\xi}$ to the thermodynamic limit in the low-temperature regime achieves close agreement with the three-loop perturbative calculations. We further propose a finite-size scaling (FSS) ansatz for $\ensuremath{\xi}$, demonstrating that the pseudocritical point ${\ensuremath{\beta}}_{L}$ diverges logarithmically with LLL. The thermodynamic and finite-size scaling behaviors of the magnetic susceptibility $\ensuremath{\chi}$ are also investigated and corroborate the predictions of asymptotic freedom. Our paper provides solid evidence for asymptotic freedom in the 2D Heisenberg model and advances understanding of finite-size scaling in such systems.
We consider a helical spin liquid system which shows majorana fermion modes at the edge. The interaction between the quasiparticles in this system induces phase transition, Majorana-Ising transition. We comply the density matrix renormalization group method to study this phase transition for the entire regime of the parameter space. We observe the presence of topological quantum phase transition for repulsive interaction, however this phase is more stable for the attractive interaction. The length scale dependent study shows many new and important results and we show explicitly that the major contribution to the excitation comes from the edge of the system when the system is in the topological state. We also show the dependence of Majorana localization length for various values of chemical potential.
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