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arXiv:cond-mat/0006503 · Journal of Physics Condensed Matter

Fractons and Luttinger liquids

Wellington da Cruz

Published Oct 18, 2000• 3 citations
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Abstract

We consider the concept of fractons as particles or quasiparticles which obey a specific fractal statistics in connection with a one-dimensional Luttinger liquid theory. We obtain a dual statistics parameter = ν + 1 which is identified with the controlling parameter e -2φ of the Luttinger model. In this way, a bosonic system characterized by a fractal index i f [ h ] = i f [2] = 1 is considered in a conformal field theory with central charge c [ν = 0] = 1 = i f [2] with a compactified radius R = 1/ 1/2 = 1. Thus, we have a mapping of a bosonic theory to a fermionic one and vice versa, i.e. the duality symmetry = 3- h of the universal class h of fractons defined in the interval 1< h <2 is satisfied.

DOI
10.1088/0953-8984/12/44/101

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We consider the concept of fractons as particles or quasiparticles which obey a specific fractal statistics in connection with a one-dimensional Luttinger liquid theory. We obtain a dual statistics parameter = ν + 1 which is identified with the controlling parameter e -2φ of the Luttinger model. In this way, a bosonic system characterized by a fractal index i f [ h ] = i f [2] = 1 is considered in a conformal field theory with central charge c [ν = 0] = 1 = i f [2] with a compactified radius R = 1/ 1/2 = 1. Thus, we have a mapping of a bosonic theory to a fermionic one and vice versa, i.e. the duality symmetry = 3- h of the universal class h of fractons defined in the interval 1< h <2 is satisfied.
We consider the quantum phase transitions of fractons in correspondence with the quantum phase transitions of the fractional quantum Hall effect-FQHE. We have that the Hall states can be modelled by fractons, known as charge-flux systems which satisfy a fractal distribution function associated with a fractal von Neumann entropy. In our formulation, the universality classes of the fractional quantum Hall transitions, are considered as fractal sets of dual topological quantum numbers filling factors labelled by the Hausdorff dimension hhh (1<h<21 < h < 21<h<2) of the quantum paths of fractons. In this way we have defined, associated to these universality classes, a scaling exponent as κ=1/hκ=1/hκ=1/h, such that when hhh runs into its interval of definition, we obtain $ 1 \gtrsim κ\gtrsim 0.5 $. The behavior of this scaling exponent, topological in character, is in agreement with some experimental values claimed in the literature. Thus, according to our approach we have a fractal scaling theory for the FQHE which distinguishes diverse universality classes for the fractional quantum Hall transitions.
We achieve a bosonization of one-dimensional ideal gas of particles obeying exclusion statistics $\ensuremath{\lambda}$ (so called $\ensuremath{\lambda}$ exclusons) at low temperatures, resulting in a new variant of c=1c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1c=1 conformal field theory with compactified radius $R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1/\ensuremath{\lambda}}$. These ideal excluson gases exactly reproduce the low- TTT critical properties of Luttinger liquids, so they can be used to characterize the fixed points of the latter. Generalized ideal gases with mutual statistics and nonideal gases with Luttinger-type interactions have also similar low- TTT behavior, controlled by an effective statistics varying in a fixed-point line.
We introduce the notion of entanglement measure for the universal classes of fractons as an entanglement between ocuppation-numbers of fractons in the lowest Landau levels and the rest of the many-body system of particles. This definition came as an entropy of the probability distribution {\it à la} Shannon. Fractons are charge-flux systems classified in universal classes of particles or quasiparticles labelled by a fractal or Hausdorff dimension defined within the interval 1<h<21 < h < 21<h<2 and associated with the fractal quantum curves of such objects. They carry rational or irrational values of spin and the spin-statistics connection takes place in this fractal approach to the fractional spin particles. We take into account the fractal von Neumann entropy associated with the fractal distribution function which each universal class of fractons satisfies. We consider the fractional quantum Hall effect-FQHE given that fractons can model Hall states. According to our formulation entanglement between occupaton-numbers in this context increases with the universality classes of the quantum Hall transitions considered as fractal sets of dual topological quantum numbers filling factors. We verify that the Hall states have stronger entanglement between ocuppation-numbers and so we can consider this resource for fracton quantum computing.
We discuss how one-dimensional interacting fermion systems, which in the low-energy approximation are described by Luttinger liquid theory, can be reformulated as systems of weakly interacting particles with fractional charge and statistics. Our approach is to use Landau's phenomenological approach to Fermi liquid theory, where the quasiparticles are interpreted as adiabatically dressed fermions. In an earlier publication, the local charge carried by these excitations has been shown to be fractional. We focus here on the statistics of the quasiparticles and show that by a change of momentum variables the Landau parameters of the generalized Fermi fluid can be transformed to zero. This change in interaction is compensated by a change of the entropy function, which is consistent with the interpretation of the quasiparticles as satisfying generalized exclusion statistics.
We show that one-dimensional quantum systems with gapless degrees of freedom and open boundary conditions form a universality class of quantum critical behavior, which we propose to call ``bounded Luttinger liquids.'' They share the following properties with ordinary (periodic) Luttinger liquids: the absence of fermionic quasiparticle excitations, charge-spin separation, and anomalous power-law correlations with exponents whose scaling relations are parametrized by a single coupling constant per degree of freedom, ${K}_{\ensuremath{\nu}}.$ The values of ${K}_{\ensuremath{\nu}}$ are independent of boundary conditions, but the representation of the critical exponents in terms of these ${K}_{\ensuremath{\nu}}'\mathrm{s}$ depends on boundary conditions. We illustrate these scaling relations by exploring general rules for boundary critical exponents derived earlier using the Bethe ansatz solution of the one-dimensional Hubbard model together with boundary conformal field theory, and the theory of Luttinger liquids in finite-size systems. We apply this theory to the photoemission properties of the organic conductors (TMTSF)2X,(\mathrm{TMTSF}{)}_{2}X,(TMTSF)2​X, where TMTSF is tetramethyltetraselenafulvalene, and X=ClO9,X={\mathrm{ClO}}_{9},X=ClO9​, PF6,{\mathrm{PF}}_{6},PF6​, ReO4,\mathrm{Re}{\mathrm{O}}_{4},ReO4​, and discuss to what extent the assumption of finite strands with open boundaries at the sample surface can reconcile the experimental results with independent information on the Luttinger-liquid state in these materials.
We explore the lifetime of excitations in a dispersive Luttinger liquid. We perform a bosonization supplemented by a sequence of unitary transformations that allows us to treat the problem in terms of weakly interacting quasiparticles. The relaxation described by the resulting Hamiltonian is analyzed by bosonic and (after a refermionization) by fermionic perturbation theory. We show that the fermionic and bosonic formulations of the problem exhibit a remarkable strong-weak coupling duality. Specifically, the fermionic theory is characterized by a dimensionless coupling constant $\ensuremath{\lambda}={m}^{*}{l}^{2}T$ and the bosonic theory by ${\ensuremath{\lambda}}^{\ensuremath{-}1}$, where 1/m∗1/{m}^{*}1/m∗ and lll characterize the curvature of the fermionic and bosonic spectra, respectively, and TTT is the temperature.