Charge fractionalization in quantum wires

Abstract

Although the unit of charge in nature is a fundamental constant, the charge of individual quasiparticles in some low-dimensional systems may be fractionalized. Quantum one-dimensional (1D) systems, for instance, are theoretically predicted to carry charge in units smaller than the electron charge e. Unlike 2D systems, the charge of these excitations is not quantized and depends directly on the strength of the Coulomb interactions. For example, in a 1D system with momentum conservation, it is predicted that the charge of a unidirectional electron that is injected into the wire decomposes into right-and left-moving charge excitations carrying fractional charges f 0 e and (1 − f 0 )e respectively 1,2 . f 0 approaches unity for non-interacting electrons and is less than one for repulsive interactions. Here, we provide the first experimental evidence for charge fractionalization in one dimension. Unidirectional electrons are injected at the bulk of a wire and the imbalance in the currents detected at two drains on opposite sides of the injection region is used to determine f 0 . Our results elucidate further 3,4 the collective nature of electrons in one dimension.Charge fractionalization in one dimension is already predicted for the spinless Luttinger model 1,2 . The charge fraction f 0 is given bywhere g c is the Luttinger-liquid interaction parameter. For a galilean invariant system, g c = v F /v c , where v F is the bare Fermi velocity and v c is the velocity of charge excitations. Roughly,where U is the Coulomb interaction energy and ε F is the Fermi energy. In spinfull one-dimensional (1D) systems, charge fractionalization occurs in addition to spin-charge separation, which has been recently confirmed by spectroscopy and tunnelling experiments [4][5][6] .Observing interaction effects in 1D systems using transport experiments is a considerable challenge. For example, the d.c. twoterminal conductance with ideal contacts is universal and given by G = G 0 ≡ 2e 2 /h, independent of interactions 7-11 . Shot-noise measurements have been useful in detecting fractional charge in 2D systems [12][13][14] . However, low-frequency shot-noise measurements in an ideal wire would only reveal the physics of the Fermi-liquid contacts, remaining insensitive to fractionalization 15 . Although both noise and conductance should reveal interaction effects at frequencies exceeding v F /g c L ∼ 10 10 Hz, where the excitation wavelength is shorter than the wire segment [16][17][18] , these frequencies are difficult to explore experimentally at low temperatures.Initial experimental indication of electron fractionalization in one dimension is provided by angle-resolved photo-emission spectroscopy measurements on stripe-ordered cuprate materials 5 . Recent theoretical studies have proposed transport experiments aimed at detecting the same physics in quantum wires. Generally, these involve the realization of multi-terminal geometries, including: (1) local injection of electrons into a wire, where high-frequency noise correlations are expecte...

Universality Relations in Non-solvable Quantum Spin Chains

“…A similar calculation can be done for Z (3) h ; in fact, in this case, there is no term corresponding to the second graph in Fig. 1, while the contribution corresponding to the first one, with the external fermion propagators of index ω, is given by…”

“…Realistic models are very difficult to study and most of the theoretical predictions for such systems (for some recent experiments see [3]) are based on a number of conjectures, whose mathematical proof is quite hard.…”

“…The parameter ν is chosen as a function of λ and p F , so that the singularity of the Fourier transform of the two-point function corresponding to H is fixed at k = (±p F , 0); the first equation in (23) gives the value of h corresponding, in the model (3), to the chosen value of p F . On the contrary, the parameter δ is an unknown function of λ and p F , whose value is determined by requiring that, in the renormalization group analysis, the corresponding marginal term flows to 0; this implies that v s is the sound velocity even for the full Hamiltonian H. It is well known that the correlations of the quantum spin chain can be derived by the following Grassmann integral, see [12]:…”

“…Hence, we can analyze (89) as we did for (63), but now we have only one r.c., to be called λ j , and three renormalization constants, Z j , Z (3) j and Z (3) j , taking the place of Z j , Z (1) j and Z (1) j , respectively. It follows that λ j → λ −∞ , as j → −∞, with λ −∞ an analytic function of λ 0 , such that λ −∞ = λ 0 + O( λ 2 0 ).…”

“…In order to complete the proof of (43), we use the representation of S (3) x S (3) 0 T , given in [12], eq. (1.13), that is…”