Localization, Dirac Fermions and Onsager Universality

Disordered systems exhibiting exponential localization are mapped to anisotropic spin chains with localization length being related to the anisotropy of the spin model. This relates localization phenomenon in fermions to the rotational symmetry breaking in the critical spin chains. One of the intriguing consequence is that the statement of Onsager universality in spin chains implies universality of the localized fermions where the fluctuations in localized wave functions are universal. We further show that the fluctuations about localized nonrelativistic fermions describe relativistic fermions. This provides a new approach to understand the absence of localization in disordered Dirac fermions. We investigate how disorder affects well known universality of the spin chains by examining the multifractal exponents. Finally, we examine the effects of correlations on the localization characteristics of relativistic fermions. PACS. PACS-key Harper equation, localization – PACS-key Dirac fermions Two-dimensional Ising model is one of the few examples of exactly solvable many body systems.[2] The model exhibits a phase transition at finite temperature characterized by universal exponents defining a universality class, the Onsager universality which describes the phase transitions for anisotropic XY models. Interestingly, the Onsager universality also describes a quantum phase transition at zero temperature, driven by quantum fluctuations, of one-dimensional quantum anisotropic XY spin chains in a transverse field.[3] These quantum models belong to a small family of integrable Hamiltonians that have attracted both theoreticians as well as experimentalists. Heart of integrability of these many body quantum spin problem is a mapping between spin and fermions which relates interacting XY spin chain with O(1) symmetry to the fermion Hamiltonian that are quadratic in fermions. The spin-fermion correspondence has proven to be extremely important also for the case of disordered magnetic field as the quadratic fermion Hamiltonian can be numerically diagonalized with extreme precision.[5] Recent studies have shown that the a large variety of disordered quantum chains with O(1) spin symmetry are still described by the Onsager universality.[4,5] In this paper, we show a new type of spin-fermion mapping where disordered fermions exhibiting exponential localization are shown to be related to anisotropic spin chain in a disordered magnetic field at the onset to long range order(LRO). This correspondence, valid exclusively for exponentially localized systems, provides new insight into various issues relevant to disordered fermion as well as spin models. In particular we exploit well established universality hypothesis of spin systems to make important statements about fermion problems. Firstly, the root of recently observed universality in localized fermions[10,11] is traced to the Onsager universality of the spin systems. Another interesting result is a correspondence between the relativistic and the nonrelativistic fermions in the presence of disorder. It is shown that the relativistic fermions can be viewed as the fluctuations in the exponentially localized solutions of the nonrelativistic fermions. This provides a new approach for understanding the absence of localization in disordered Dirac fermions which has been the subject of various recent studies.[12] We argue that the long range magnetic correlations provide mechanism for delocalization of relativistic fermions thus obtaining a deeper understanding of the absence of localization. In addition to obtaining intuitively appealing picture of some surprising results of disordered fermions, we also obtain a generalization of universality statement of the critical exponents for disordered spin models. Finally, we examine how the correlations affect the localization characteristics and show the possibility of delocalization of relativistic fermions analogous to the corresponding nonrelativistic case. Although the setting we describe is quite general in the context of disordered systems, for concreteness we will consider quasiperiodic disorder where the lattice problem for nonrelativistic fermion is described by the Harper equation,[13] ψn−1 + ψn+1 + 2λVnψn = Eψn. (1) Here Vn = cos(θn) where θn = (2πσgn + φ). The σg is an irrational number describing competing length scales in the problem and φ is a constant phase factor. Harper equation, in one-band approximation describes Bloch electrons in a magnetic field. This problem frequently arises 2 Indubala I. Satija[1]: Localization, Dirac Fermions and Onsager Universality in many different physical contexts, every time emerging with a new face to describe another physical application. The problem is solvable by Bethe-ansatz[6]: there is an algebraic Bethe-ansatz equation for the spectrum. It was shown that at some special points in the spectrum, e.g at the mid-band points, the Hamiltonian in certain gauge can be written as a linear combination of generators of a quantum group called Uq(sl2). It also describes some properties of the integer quantum Hall effect: it showed that Kubo-Greenwood formula for the conductance of any filled isolated band is an expression for a topological invariant, and is an integer multiple of e/h. It was further shown by Avron et al[7] that it is a topological invariant that defines the first Chern class of the mapping of the Brillouin zone( a two-dimensional torus) onto a complex projective space of the wave functions. In contrast to usual Anderson problem describing localized particle in a random potential, the Harper equation exhibits localization-delocalization transition provided σg is an irrational number with good Diophantine properties (ie. badly approximated by rational numbers). This transition at λ = 1 has been characterized with singular continuous states and spectra. Richness and complexity of the critical point describing localization transition has been studied in great detail by various renormalization group approaches.[8,9] Recently, it was shown that multifractal characteristics continue to exist beyond the critical point[10] throughout the localized phase. This hidden complexity of the localized phase is brought to light after one factors out the exponentially decaying envelope. The localized wave functions with inverse localization length ξ = log(λ) is rewritten as,[10]