Random transverse and longitudinal field Ising chains

TL;DR

This paper investigates the low-energy behavior of a random-field Ising chain with both transverse and longitudinal disorder using strong disorder renormalization group techniques, revealing fixed points and critical exponents.

Contribution

It provides a detailed analysis of the phase diagram and critical properties of the model with combined random transverse and longitudinal fields.

Findings

Identifies trivial and non-trivial fixed points depending on the absence of fields.

Shows the correlation length diverges with an exponent    at the critical point.

RG trajectories are attracted to fixed points determined by the presence or absence of disorder types.

Abstract

Motivated by experimental results on compounds like ${\rm LiHo}_x{\rm Y}_{1-x}{\rm F}_4$, we consider an Ising chain with random bonds in the simultaneous presence of random transverse and longitudinal fields. We study the low-energy properties of the model at zero temperature by the strong disorder renormalization group (SDRG) method.In the absence of random longitudinal fields, the model showcases a trivial quantum-ordered and quantum-disordered fixed-point and a non-trivial infinite disorder critical point. In the absence of random transverse fields, the behavior is dictated by the classical random-field Ising fixed-point. In the simultaneous presence of both a longitudinal and transverse random field, the RG trajectories are attracted to one of the two disordered fixed-points and the relevant scaling direction at the infinite disorder fixed-point is along the separatrix, where the correlation-length is shown to diverge with an exponent $\nu_h \approx 1$.