The real-space renormalisation group method can be applied to the Chalker–Coddington model of the quantum Hall transition to provide a convenient numerical estimation of the localisation critical exponent, ν . Previous such studies found ν ∼ 2 . 39 which falls considerably short of the current best estimates by transfer matrix ( ν = 2 . 593 + 0 . 005 − 0 . 006 ) and exact-diagonalisation studies ( ν = 2 . 58 ( 3 ) ). By increasing the amount of data 500 fold we can now measure closer to the critical point and find an improved estimate ν = 2 . 51 + 0 . 11 − 0 . 11 . This deviates only ∼ 3% from the previous two values and is already better than the ∼ 7% accuracy of the classical small-cell renormalisation approach from which our method is adapted. We also study a previously proposed mixing of the Chalker–Coddington model with a classical scattering model which is meant to provide a route to understanding why experimental estimates give a lower ν ∼ 2 . 3 . Upon implementing this mixing into our RG unit, we find only further increases to the value of ν .
The real-space renormalisation group method can be applied to the Chalker–Coddington model of the quantum Hall transition to provide a convenient numerical estimation of the localisation critical exponent, ν . Previous such studies found ν ∼ 2 . 39 which falls considerably short of the current best estimates by transfer matrix ( ν = 2 . 593 + 0 . 005 − 0 . 006 ) and exact-diagonalisation studies ( ν = 2 . 58 ( 3 ) ). By increasing the amount of data 500 fold we can now measure closer to the critical point and find an improved estimate ν = 2 . 51 + 0 . 11 − 0 . 11 . This deviates only ∼ 3% from the previous two values and is already better than the ∼ 7% accuracy of the classical small-cell renormalisation approach from which our method is adapted. We also study a previously proposed mixing of the Chalker–Coddington model with a classical scattering model which is meant to provide a route to understanding why experimental estimates give a lower ν ∼ 2 . 3 . Upon implementing this mixing into our RG unit, we find only further increases to the value of ν .
Introduction The real space renormalisation group has a wide application in the study of classical critical phenomena. For quantum systems, several versions of the real space renormalisation group have been proposed (Jullien, 1981; Pfeuty et al ., 1982; dos Santos, 1982). The real space RG is the most direct way of implementing the scaling ideas presented in Chapter 1. The operation of elimination of degrees of freedom as the system is reduced to blocks of smaller sizes under a change of length scale is directly and intuitively represented as mathematical equations. In particular, the block method has been used to study different problems of strongly correlated systems (Pfeuty et al ., 1982) and spin chains (White, 1992, 1993) with different degrees of sophistication. We show here in detail a simple form of this technique applied to the case of the one-dimensional Ising model in a transverse field. The purpose is to illustrate the ideas introduced earlier in Chapter 1 with a concrete example where a quantum critical point is associated with an unstable fixed point. Also, we show how the stable attractors or fixed points can characterise the different phases and how the critical exponents can be obtained. Although the actual values for the quantum critical point and critical exponents that we obtain using a small block are not exact, a procedure is outlined which can be used to improve these results. In this sense, the block method presented here represents a controlled approximation, which can be systematically improved to yield in the limit of very large blocks essentially the correct results. When compared to the k -space version of the RG (to be presented in the next two chapters), the real space version turns out to be in general more difficult to control, in the sense of producing a systematic improvement of the results. On the other hand, this method yields without much effort the attractive fixed points of the different phases, allowing for their characterisation. This turns out to be useful since the nature of the phase transition, whether first or second order, can be determined by exponents calculated in the attractor of this phase (Nienhuis and Naunberg, 1975).
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.
The quantum Hall effect is one of the most extensively studied topological effects in solid‐state physics. The transitions between different quantum Hall states exhibit critical phenomena described by universal critical exponents. Numerous numerical finite size scaling studies have focused on the critical exponent ν for the correlation length. In such studies, it is important to take proper account of irrelevant corrections to scaling. Recently, Dresselhaus et al. proposed a new scaling ansatz, which they applied to the two terminal conductance of the Chalker–Coddington model. Herein, their proposal is applied to the previously reported data for the Lyapunov exponents of that model using both polynomial fitting and Gaussian process fitting.
We develop a novel real-space renormalization group (RG) scheme which accurately estimates correlation length exponent ν near criticality of higher-dimensional quantum Ising and Potts models in a transverse field. Our method is remarkably simple (often analytical), grouping only a few spins into a block spin so that renormalized Hamiltonian has a closed form. A previous difficulty of spatial anisotropy and unwanted terms is avoided by incorporating rotational invariance and internal Zq symmetries of the Hamiltonian. By applying this scheme to the (2+1)-dim Ising model on a triangular lattice and solving an analytical RG equation, we obtain ν≈0.6300. This value is within statistical errors of the current best Monte-Carlo result, 25th-order high-temperature series expansions, ϕ4-theory estimation which considers up to seven-loop corrections and experiments performed in low-Earth orbits. We also apply the scheme to higher-dimensional Potts models for which ordinary Monte-Carlo methods are not effective due to strong hysteresis and suppression of quantum fluctuation in a weak first-order phase transition.
Abstract We focus on two real-space renormalization-group (RG) methods recently proposed for a hierarchical model of a spin glass: a sample-by-sample method, in which the RG transformation is performed separately on each disorder sample, and an ensemble RG (ERG) method (Angelini M C, Parisi G and Ricci-Tersenghi F 2013 Ensemble renormalization group for disordered systems Phys. Rev . B 87 134201) in which the transformation is based on an average over samples. Above the upper critical dimension, the sample-by-sample method yields the correct mean-field value for the critical exponent related to the divergence of the correlation length, while it does not predict the correct qualitative behavior of below the upper critical dimension. On the other hand, the ERG procedure has been claimed to predict the correct behavior of both above and below the upper critical dimension. Here, we straighten out the reasons for the discrepancy between the two methods above, by demonstrating that the ERG method predicts a marginally stable critical fixed point, thus implying a prediction for the critical exponent given by . This prediction disagrees, on a qualitative and quantitative level, both with the mean-field value of above the critical dimension, and with numerical estimates of below the upper critical dimension. Therefore, our results show that finding a real-space RG method for spin glasses which yields the correct prediction for universal quantities below the upper critical dimension is still an open problem, for which our analysis may provide some general guidance for future studies.