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ランダム量子スピン系の実空間くりこみ群による解析

涼二 宮崎, Ryoji Miyazaki

Published Mar 1, 2014• 0 citations
OpenAlex

Abstract

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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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.
The transverse-field Ising models with random exchange interactions in finite dimensions are investigated by means of a real-space renormalization-group method. The scheme yields the exact values of the critical point and critical exponent $\ensuremath{\nu}$ in one dimension and some previous results in the case of random ferromagnetic interactions are reproduced in two and three dimensions. We apply the scheme to spin glasses in transverse fields in two and three dimensions, which have not been analyzed very extensively. The phase diagrams and the critical exponent $\ensuremath{\nu}$ are obtained and evidence for the existence of an infinite-randomness fixed point in these models is found.
The two- and three-dimensional transverse-field Ising models with ferromagnetic exchange interactions are analyzed by means of the real-space renormalization-group method. The basic strategy is a generalization of a method developed for the one-dimensional case, which exploits the exact invariance of the model under renormalization and is known to give the exact values of the critical point and critical exponent ν. The resulting values of the critical exponent ν in two and three dimensions are in good agreement with those for the classical Ising model in three and four dimensions. To the best of our knowledge, this is the first example in which a real-space renormalization group on (2+1)- and (3+1)-dimensional Bravais lattices yields accurate estimates of the critical exponents.
We apply a real-space block renormalization group approach to study the critical properties of the random transverse-field Ising spin chain with multispin interactions. First, we recover the known properties of the traditional model with two-spin interactions by applying the renormalization approach for the arbitrary size of the block. For the model with three-spin couplings, we calculate the critical point and demonstrate that the phase transition is controlled by an infinite disorder fixed point. We have determined the typical correlation-length critical exponent, which seems to be different from that of the random transverse Ising chain with nearest-neighbor couplings. Thus, this model represents a new infinite disorder universality class.
The infinite-disorder fixed point of the random transverse-field Ising model is expected to control the critical behavior of a large class of random quantum and stochastic systems having an order parameter with discrete symmetry. Here we study the model on the square lattice with a very efficient numerical implementation of the strong disorder renormalization group method, which makes us possible to treat finite samples of linear size up to L=2048L=2048L=2048. We have calculated sample dependent pseudocritical points and studied their distribution, which is found to be characterized by the same shift and width exponent: $\ensuremath{\nu}=1.24(2)$. For different types of disorder the infinite-disorder fixed point is shown to be characterized by the same set of critical exponents, for which we have obtained improved estimates: x=0.982(15)x=0.982(15)x=0.982(15) and $\ensuremath{\psi}=0.48(2)$. We have also studied the scaling behavior of the magnetization in the vicinity of the critical point as well as dynamical scaling in the ordered and disordered Griffiths phases.
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.
The random transverse-field Ising ferromagnet (RTFIF) is a highly disordered quantum system which contains randomness in the coupling strengths as well as in the transverse-field strengths. In one dimension, the critical properties are governed by an infinite-randomness fixed point (IRFP), and renormalization-group studies argue that the two-dimensional (2D) model is also governed by an IRFP. However, even the location of the critical point remains unsettled among quantum Monte Carlo (QMC) studies. In this work, we perform extensive QMC simulations to locate the quantum critical point and attempt a finite-size scaling analysis to observe the critical behavior. We estimate the critical field strength of the 2D RTFIF as ${\mathrm{\ensuremath{\Gamma}}}_{c}=7.52(2)$, together with critical exponents such as $\ensuremath{\beta}=1.5(3), \ensuremath{\nu}=1.6(3)$, and z=3.3(3)z=3.3(3)z=3.3(3) or $\ensuremath{\psi}=0.50(3)$. We have also considered the McCoy-Wu model, which has randomness in the ferromagnetic coupling strengths but not in the transverse-field strength. Our QMC calculation shows that the critical behavior of the 2D McCoy-Wu model is closer to that of the 2D transverse-field Ising spin glass than to that of the 2D RTFIF. These numerical findings enhance our understanding of disordered 2D quantum systems.