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Journal of Physics A Mathematical and General

Static and dynamic aspects of disorder lines

Thomas Garel, J. C. Niel, Henri Orland, Michael Schick C

Published Mar 21, 1991• 3 citations
Publisher

Abstract

Using one-dimensional models, the authors characterize the existence of a disorder line (point) in a renormalization group procedure. This procedure is defined in terms of the eigenvalues of the transfer matrix. There are various possibilities for the crossing of subdominant eigenvalues; this study therefore encompasses usual disorder lines (where the spin-spin correlation function changes), as well as other types of disorder lines (where, for instance, the energy-energy correlation function changes). In the usual case, the disorder line shows up as a separatrix between two regions of the flow diagram attracted to different paramagnetic sinks. For this case, they also consider, in the framework of the random phase approximation, the dynamical signature of a disorder line: they show that it can be experimentally detected, especially for disorder lines (points) of the second kind.

DOI
10.1088/0305-4470/24/6/017

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Using one-dimensional models, the authors characterize the existence of a disorder line (point) in a renormalization group procedure. This procedure is defined in terms of the eigenvalues of the transfer matrix. There are various possibilities for the crossing of subdominant eigenvalues; this study therefore encompasses usual disorder lines (where the spin-spin correlation function changes), as well as other types of disorder lines (where, for instance, the energy-energy correlation function changes). In the usual case, the disorder line shows up as a separatrix between two regions of the flow diagram attracted to different paramagnetic sinks. For this case, they also consider, in the framework of the random phase approximation, the dynamical signature of a disorder line: they show that it can be experimentally detected, especially for disorder lines (points) of the second kind.
We show analytically how the crossing of a disorder line in a simple one-dimensional model coincides with a qualitative change of behaviour (monotonous to nonmonotonous) in the renormalization group flow. Such a change is found in Migdal-Kadanoff recursion relations of a family of ferromagnetic two-dimensional XY models. We therefore infer the existence of a disorder line in these models. For the usual XY model, the disorder temperature corresponds to the onset of short-distance pairing of vortices. It may coincide with the dynamical transition temperature recently found by Golinelli and Derrida.
We formulate a strong-disorder renormalization-group (SDRG) approach to study the $\ensuremath{\beta}$ function of the tight-binding model in one dimension with both diagonal and off-diagonal disorder for states at the band center. We show that the SDRG method, when used to compute transport properties, yields exact results since it is identical to the transfer matrix method. The $\ensuremath{\beta}$ function is shown to be universal when only off-diagonal disorder is present even though single-parameter scaling is known to be violated. A different single-parameter scaling theory is formulated for this particular (particle-hole symmetric) case. Upon breaking particle-hole symmetry (by adding diagonal disorder), the $\ensuremath{\beta}$ function is shown to crossover from the universal behavior of the particle-hole symmetric case to the conventional nonuniversal one in agreement with the two-parameter scaling theory. We finally draw an analogy with the random transverse-field Ising chain in the paramagnetic phase. The particle-hole symmetric case corresponds to the critical point of the quantum Ising model, while the generic case corresponds to the Griffiths paramagnetic phase.
We numerically study the phase diagram and critical properties of the two-dimensional disordered O(n) loop model by using the transfer matrix and the worm Monte Carlo methods. The renormalization group flow is extracted from the landscape of the effective central charge obtained by the transfer matrix method based on the Zamolodchikov's C-theorem. We find a line of random fixed points (FPs) for nc<n<1n_c < n <1nc​<n<1, with nc∼0.5n_c \sim 0.5nc​∼0.5, for which the central charge and critical exponents agree well with the results of the 1−n1-n1−n perturbative expansion. Furthermore, for n>ncn> n_cn>nc​, we find a line of multicritical FPs at strong randomness. The FP at n=1n=1n=1 has c=0.4612(4)c=0.4612(4)c=0.4612(4), which suggests that it belongs to the universality class of the Nishimori point in the ±J\pm J±J random-bond Ising model. For n>2n>2n>2, we find another critical line that connects the hard-hexagon FP in the pure model to a finite-randomness zero-temperature FP.
Abstract We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points for O(N )-symmetric models with quenched disorder in two dimensions. Random fixed points are characterized by two disorder parameters: a modulus that vanishes when approaching the pure case, and a phase angle. The critical lines fall into three classes depending on the values of the disorder modulus. Besides the class corresponding to the pure case, a second class has maximal value of the disorder modulus and includes Nishimori-like multicritical points as well as zero temperature fixed points. The third class contains critical lines that interpolate, as N varies, between the first two classes. For positive N , it contains a single line of infrared fixed points spanning the values of N from 2−1 \sqrt{2}-1 2​−1 2 − 1 to 1. The symmetry sector of the energy density operator is superuniversal (i.e. N -independent) along this line. For N = 2 a line of fixed points exists only in the pure case, but accounts also for the Berezinskii-Kosterlitz-Thouless phase observed in presence of disorder.
Critical properties of quantum spin chains with varying degrees of disorder are studied at zero temperature by analytical and extensive density matrix renormalization methods. Generally the phase diagram is found to contain three phases. The weak disorder regime, where the critical behavior is controlled by the fixed points of the pure system, and the strong disorder regime, which is attracted by an infinite randomness fixed point, are separated by an intermediate disorder regime, where dynamical scaling is anisotropic and the static and dynamical exponents are disorder dependent.
In this paper we promote the idea of quantum critical lines (inter alia surfaces) as opposed to points. A quantum critical line is obtained when criticality at zero temperature is extended over a continuum in a one-dimensional line. We base our ideas on a simple but exactly solved model introduced by one of the authors involving a one-dimensional quantum transverse field Ising model with added three-spin interaction. While many of the ideas are quite general, there are other aspects that are not. In particular, a line of criticality with continuously varying exponents is not captured. However, the exact solvability of the model gives us considerable confidence in our results. Although the pure system is analytically exactly solved, the disorder case requires numerical analyses based on exact computation of the correlation function in the Pfaffian representation. The disorder case leads to a dynamic structure factor as a function of frequency and wave vector. We expect that the model is experimentally realizable and perhaps many other similar models will be found to explore quantum critical lines.