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Register a new userCritical behavior of the quasi-periodic quantum Ising chain
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The interplay of correlated spatial modulation and symmetry breaking leads to quantum critical phenomena intermediate between those of the clean and randomly disordered cases. By performing a detailed analytic and numerical case study of the quasi-periodically (QP) modulated transverse field Ising chain, we provide evidence for the conjectures of Ref.~cite{crowley2018quasi} regarding the QP-Ising universality class. In the generic case, we confirm that the logarithmic wandering coefficient $w$ governs both the macroscopic critical exponents and the energy-dependent localisation length of the critical excitations. However, for special values of the phase difference $Delta$ between the exchange and transverse field couplings, the QP-Ising transition has different properties. For $Delta=0$, a generalised Aubry-Andre duality prevents the finite energy excitations from localising despite the presence of logarithmic wandering. For $Delta$ such that the fields and couplings are related by a lattice shift, the wandering coefficient $w$ vanishes. Nonetheless, the presence of small couplings leads to non-trivial exponents and localised excitations. Our results add to the rich menagerie of quantum Ising transitions in the presence of spatial modulation.
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Unlike random potentials, quasi-periodic modulation can induce localisation-delocalisation transitions in one dimension. In this article, we analyse the implications of this for symmetry breaking in the quasi-periodically modulated quantum Ising chain. Although weak modulation is irrelevant, strong modulation induces new ferromagnetic and paramagnetic phases which are fully localised and gapless. The quasi-periodic potential and localised excitations lead to quantum criticality that is intermediate to that of the clean and randomly disordered models with exponents of $ u=1^{+}$, and $zapprox 1.9$, $Delta_sigma approx 0.16$, $Delta_gammaapprox 0.63$ (up to logarithmic corrections). Technically, the clean Ising transition is destabilized by logarithmic wandering of the local reduced couplings. We conjecture that the wandering coefficient $w$ controls the universality class of the quasi-periodic transition and show its stability to smooth perturbations that preserve the quasi-periodic structure of the model.
We study critical behavior of the diluted 2D Ising model in the presence of disorder correlations which decay algebraically with distance as $sim r^{-a}$. Mapping the problem onto 2D Dirac fermions with correlated disorder we calculate the critical properties using renormalization group up to two-loop order. We show that beside the Gaussian fixed point the flow equations have a non trivial fixed point which is stable for $0.995<a<2$ and is characterized by the correlation length exponent $ u= 2/a + O((2-a)^3)$. Using bosonization, we also calculate the averaged square of the spin-spin correlation function and find the corresponding critical exponent $eta_2=1/2-(2-a)/4+O((2-a)^2)$.
A novel order parameter $Phi$ for spin glasses is defined based on topological criteria and with a clear physical interpretation. $Phi$ is first investigated for well known magnetic systems and then applied to the Edwards-Anderson $pm J$ model on a square lattice, comparing its properties with the usual $q$ order parameter. Finite size scaling procedures are performed. Results and analyses based on $Phi$ confirm a zero temperature phase transition and allow to identify the low temperature phase. The advantages of $Phi$ are brought out and its physical meaning is established.
We consider the paramagnetic phase of the random transverse-field Ising spin chain and study the dynamical properties by numerical methods and scaling considerations. We extend our previous work [Phys. Rev. B 57, 11404 (1998)] to new quantities, such as the non-linear susceptibility, higher excitations and the energy-density autocorrelation function. We show that in the Griffiths phase all the above quantities exhibit power-law singularities and the corresponding critical exponents, which vary with the distance from the critical point, can be related to the dynamical exponent z, the latter being the positive root of [(J/h)^{1/z}]_av=1. Particularly, whereas the average spin autocorrelation function in imaginary time decays as [G]_av(t)~t^{-1/z}, the average energy-density autocorrelations decay with another exponent as [G^e]_av(t)~t^{-2-1/z}.
We use finite size scaling to study Ising spin glasses in two spatial dimensions. The issue of universality is addressed by comparing discrete and continuous probability distributions for the quenched random couplings. The sophisticated temperature dependency of the scaling fields is identified as the major obstacle that has impeded a complete analysis. Once temperature is relinquished in favor of the correlation length as the basic variable, we obtain a reliable estimation of the anomalous dimension and of the thermal critical exponent. Universality among binary and Gaussian couplings is confirmed to a high numerical accuracy.
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