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arXiv:1711.04783 · Physical review. B./Physical review. B

Highly symmetric random one-dimensional spin models

Victor L. Quito, Pedro L. S. Lopes, José A. Hoyos, E. Miranda

Published Dec 6, 2019• 9 citations
arXivHTMLPDF

Abstract

The interplay of disorder and interactions is a challenging topic of condensed matter physics, where correlations are crucial and exotic phases develop. In one spatial dimension, a particularly successful method to analyze such problems is the strong-disorder renormalization group (SDRG). This method, which is asymptotically exact in the limit of large disorder, has been successfully employed in the study of several phases of random magnetic chains. Here we develop an SDRG scheme capable of providing in-depth information on a large class of strongly disordered one-dimensional magnetic chains with a global invariance under a generic continuous group. Our methodology can be applied to any Lie-algebra valued spin Hamiltonian, in any representation. As examples, we focus on the physically relevant cases of SO($N$) and Sp($N$) magnetism, showing the existence of different randomness-dominated phases. These phases display emergent SU($N$) symmetry at low energies and fall into two distinct classes, with meson-like or baryon-like characteristics. Our methodology is here explained in detail and helps to shed light on a general mechanism for symmetry emergence in disordered systems.

DOI
10.1103/physrevb.100.224407

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The interplay of disorder and interactions is a challenging topic of condensed matter physics, where correlations are crucial and exotic phases develop. In one spatial dimension, a particularly successful method to analyze such problems is the strong-disorder renormalization group (SDRG). This method, which is asymptotically exact in the limit of large disorder, has been successfully employed in the study of several phases of random magnetic chains. Here we develop an SDRG scheme capable of providing in-depth information on a large class of strongly disordered one-dimensional magnetic chains with a global invariance under a generic continuous group. Our methodology can be applied to any Lie-algebra valued spin Hamiltonian, in any representation. As examples, we focus on the physically relevant cases of SO(NNN) and Sp(NNN) magnetism, showing the existence of different randomness-dominated phases. These phases display emergent SU(NNN) symmetry at low energies and fall into two distinct classes, with meson-like or baryon-like characteristics. Our methodology is here explained in detail and helps to shed light on a general mechanism for symmetry emergence in disordered systems.
We study the low-energy physics of a broad class of time-reversal invariant and SU(2)-symmetric one-dimensional spin-SSS systems in the presence of quenched disorder via a strong-disorder renormalization-group technique. We show that, in general, there is an antiferromagnetic phase with an emergent SU(2S+1)\mathrm{SU}(2S+1)SU(2S+1) symmetry. The ground state of this phase is a random singlet state in which the singlets are formed by pairs of spins. For integer spins, there is an additional antiferromagnetic phase which does not exhibit any emergent symmetry (except for S=1)S=1)S=1). The corresponding ground state is a random singlet one but the singlets are formed mostly by trios of spins. In each case the corresponding low-energy dynamics is activated, i.e., with a formally infinite dynamical exponent, and related to distinct infinite-randomness fixed points. The phase diagram has two other phases with ferromagnetic tendencies: a disordered ferromagnetic phase and a large spin phase in which the effective disorder is asymptotically finite. In the latter case, the dynamical scaling is governed by a conventional power law with a finite dynamical exponent.
We train machine learning algorithms to infer the entanglement structure of disordered long-range interacting quantum spin chains by learning from the strong disorder renormalisation group (SDRG) method. The system consists of S=1/2S=1/2S=1/2-quantum spins coupled by antiferromagnetic power-law interactions with decay exponent ααα at random positions on a one-dimensional chain. Using SDRG as a physics-informed teacher, we compare a Random Forest classifier as a classical baseline with a graph neural network (GNN) that operates directly on the interaction graph and learns a bond-ranking rule mirroring the SDRG decimation policy. The GNN achieves a disorder-averaged pairing accuracy close to one and reproduces the entanglement entropy S(ℓ)S(\ell)S(ℓ) in excellent quantitative agreement with SDRG across all subsystem sizes and interaction exponents. RG flow heat maps confirm that the GNN learns the sequential decimation hierarchy rather than merely fitting final-state observables. Finite-temperature entanglement properties are incorporated via the SDRGX framework through a two-stage strategy, using the zero-temperature GNN to generate the RG flow and sampling thermal occupations from the canonical ensemble, yielding results in agreement with both numerical SDRGX and analytical predictions without retraining.
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.
This paper focuses on developing a new algorithm (En-DMRG) for obtaining highly excited states of disordered and strongly interacting spin systems. This line of research is stimulated by the growing interest in understanding the many-body localized (MBL) state, which emerges in disordered interacting systems at high energy densities through a disorder-driven dynamic phase transition. By studying the one-dimensional Heisenberg spin chain (with up to 72 spins) in a random field, the authors demonstrate the accuracy of the method in obtaining energy eigenstates and the corresponding statistical results of the quantum system in the MBL phase. The method presented in this work has the potential to be applicable in a variety of quantum statistical physics problems, possibly including the MBL phase in higher dimensions.
Motivated by long-range dispersal in ecological systems, we formulate and apply a general strong-disorder renormalization group (SDRG) framework to describe one-dimensional disordered contact processes with heavy-tailed (such as power law, stretched exponential, and log-normal) dispersal kernels, widely used in ecology. The focus is on the close-to-critical scaling of the order parameters, including the commonly used density, as well as the less known persistence, which is nonzero in the inactive phase. Our analytic and numerical results obtained by SDRG schemes at different levels of approximation reveal that the more slowly decaying dispersal kernels lead to faster-vanishing densities as the critical point is approached. The persistence, however, shows an opposite tendency: the broadening of the dispersal makes its decline sharper at the critical point, becoming discontinuous for the extreme case of power-law dispersal. The SDRG schemes presented here also describe the quantum phase transition of random transverse-field Ising chains with ferromagnetic long-range interactions, the density corresponding to the magnetization of that model.
We use a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations to study S=1/2S=1/2S=1/2 spin chains with random couplings, calculating disorder-averaged spin and dimer correlations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multispin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that cannot be generated by the SDRG scheme. In the model with multispin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties.