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arXiv:0707.0293 · Physical Review Letters

Irrational Versus Rational Charge and Statistics in Two-Dimensional Quantum Systems

Claudio Chamon, Chang-Yu Hou, R. Jackiw, Christopher Mudry, So-Young Pi, Andreas P. Schnyder

Published Mar 21, 2008• 68 citations
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Abstract

We show that quasiparticle excitations with irrational charge and irrational exchange statistics exist in tight-binding systems described, in the continuum approximation, by the Dirac equation in (2+1)-dimensional space and time. These excitations can be deconfined at zero temperature, but when they are, the charge rerationalizes to the value 1/2 and the exchange statistics to that of "quartons" (half-semions).

DOI
10.1103/physrevlett.100.110405

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We show that quasiparticle excitations with irrational charge and irrational exchange statistics exist in tight-binding systems described, in the continuum approximation, by the Dirac equation in (2+1)-dimensional space and time. These excitations can be deconfined at zero temperature, but when they are, the charge rerationalizes to the value 1/2 and the exchange statistics to that of "quartons" (half-semions).
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Elementary, low-energy excitations are examined by bosonization in one-dimensional systems with quasi-long-range order. A new, independently measurable attribute is introduced to describe such excitations. It is defined as a number www which determines how many times the phase of the order parameter winds as an excitation is transposed from far left to far right. The winding number is zero for electrons and holes with conventional quantum numbers, but it acquires a nontrivial value w=1w\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1w=1 for neutral spin- 12\frac{1}{2}21​ excitations and for spinless excitations with a unit electron charge. It may even be irrational, if the charge is irrational. Thus, these excitations are topological.
Elementary, low-energy excitations are examined by bosonization in one-dimensional systems with quasi-long-range order. A new, independently measurable attribute is introduced to describe such excitations. It is defined as a number w which determines how many times the phase of the order parameter winds as an excitation is transposed from far left to far right. The winding number is zero for electrons and holes with conventional quantum numbers, but it acquires a nontrivial value w=1 for neutral spin- (1) /(2) excitations and for spinless excitations with a unit electron charge. It may even be irrational, if the charge is irrational. Thus, these excitations are topological. {copyright} {ital 1999} {ital The American Physical Society }
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 present numerical solutions of the spectral functions of large dimension $t\text{\ensuremath{-}}J$ models with random nearest-neighbor exchange and global SU(M)\mathrm{SU}(M)SU(M) spin rotation symmetry. The solutions are obtained from the saddle-point equations of the large dimension limit, followed by the large MMM limit. The same saddle point equations also apply to the model with all-to-all and random hopping and exchange. Such a $t\text{\ensuremath{-}}J$ model realizes a deconfined critical state proposed as a model for the optimally doped cuprates. The large MMM theory involves Green's functions for fractionalized spinons and holons carrying emergent U(1) gauge charges, obeying relations similar to those of the Sachdev-Ye-Kitaev (SYK) models. The low frequency spectral functions are compared with an analytic analysis of the operator scaling dimensions with good agreement. We also compute the low frequency and temperature behavior of experimentally observable gauge-invariant observables: the electron Green's function, the local spin susceptibility and the optical conductivity, along with the temperature dependence of the d.c. resistivity. The time reparameterization soft mode (equivalent to the boundary graviton in holographically dual models of two-dimensional quantum gravity) makes important contributions to all observables and provides a linear-in-temperature contribution to the d.c. resistivity.