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arXiv:2304.10564 · Physical review. B./Physical review. B

Random geometry at an infinite-randomness fixed point

Akshat Pandey, Aditya Mahadevan, Aditya Cowsik

Published Aug 2, 2023• 2 citations
arXivHTMLPDF

Abstract

We study the low-energy physics of the critical ($2+1$)-dimensional random transverse-field Ising model. The one-dimensional version of the model is a paradigmatic example of a system governed by an infinite-randomness fixed point, for which many results on the distributions of observables are known via an asymptotically exact renormalization group (RG) approach. In two dimensions, the same RG rules have been implemented numerically and demonstrate a flow to infinite randomness. However, analytical understanding of the structure of this RG has remained elusive due to the development of geometrical structure in the graph of interacting spins. To understand the character of the fixed point, we consider the RG flow acting on a joint ensemble of graphs and couplings. We propose that the RG effectively occurs in two stages: (1) randomization of the interaction graph until it belongs to a certain ensemble of random triangulations of the plane and (2) a flow of the distributions of couplings to infinite randomness while the graph ensemble remains invariant. This picture is substantiated by a numerical RG in which one obtains a steady-state graph degree distribution and subsequently infinite-randomness scaling distributions of the couplings. Both of these aspects of the RG flow can be approximately reproduced in simplified analytical models.

DOI
10.1103/physrevb.108.064201

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We study the low-energy physics of the critical (2+12+12+1)-dimensional random transverse-field Ising model. The one-dimensional version of the model is a paradigmatic example of a system governed by an infinite-randomness fixed point, for which many results on the distributions of observables are known via an asymptotically exact renormalization group (RG) approach. In two dimensions, the same RG rules have been implemented numerically and demonstrate a flow to infinite randomness. However, analytical understanding of the structure of this RG has remained elusive due to the development of geometrical structure in the graph of interacting spins. To understand the character of the fixed point, we consider the RG flow acting on a joint ensemble of graphs and couplings. We propose that the RG effectively occurs in two stages: (1) randomization of the interaction graph until it belongs to a certain ensemble of random triangulations of the plane and (2) a flow of the distributions of couplings to infinite randomness while the graph ensemble remains invariant. This picture is substantiated by a numerical RG in which one obtains a steady-state graph degree distribution and subsequently infinite-randomness scaling distributions of the couplings. Both of these aspects of the RG flow can be approximately reproduced in simplified analytical models.
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The present thesis investigates phase transitions and critical phenomena in the random transverse-field Ising models by means of a real-space renormalization-group method.The analysis is intended to examine the presence of infinite-randomness fixed points in the models in a renormalization-group picture.Infinite-randomness fixed points are concerned with peculiar critical phenomena, which have been established in a study of the one-dimensional random transverse-field Ising model.Subsequent studies have conjectured the presence of the fixed point in some cases.The thesis mainly examines such two cases: the two-and three-dimensional Ising spin glasses in a transverse field and the one-dimensional Ising spin glass with the power-law interaction in a transverse field.The latter is not directly tackled.Instead, we introduce and analyze the hierarchical model which is expected to show similar critical phenomena.A real-space renormalization-group method is exploited in the thesis.We develop the method in a model without randomness first.The method reproduces some previously known results of transition points and critical exponents in pure and disordered cases.The spin-glass models mentioned above are investigated with the method.We find evidence of the presence of infinite-randomness fixed points in these cases as a result, which supports the conjectures.
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