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Renormalization Group and Asymptotic Spin–Charge separation for Chiral Luttinger liquids.

P. Falco, V. Mastropietro

Published Nov 16, 2011• 5 citations
OpenAlex

Abstract

The phenomenon of Spin-Charge separation in non-Fermi liquids is well understood only in certain solvable d = 1 fermionic systems. In this paper we furnish the first example of asymptotic Spin-Charge separation in a d = 1 non solvable model. This goal is achieved using Renormalization Group approach combined with Ward-Identities and Schwinger-Dyson equations, corrected by the presence of a bandwidth cut-offs. Such methods, contrary to bosonization, could be in principle applied also to lattice or higher dimensional systems. 1 Introduction and Main results In recent years the properties of non-Fermi liquids have been extensively investigated, both from experimental and theoretical point of view. In particular, one of the most spectacular feature appearing in non-Fermi liquids is the phenomenon of Spin-Charge (SC) separation, which is surely relevant for the physics

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The phenomenon of Spin-Charge separation in non-Fermi liquids is well understood only in certain solvable d = 1 fermionic systems. In this paper we furnish the first example of asymptotic Spin-Charge separation in a d = 1 non solvable model. This goal is achieved using Renormalization Group approach combined with Ward-Identities and Schwinger-Dyson equations, corrected by the presence of a bandwidth cut-offs. Such methods, contrary to bosonization, could be in principle applied also to lattice or higher dimensional systems. 1 Introduction and Main results In recent years the properties of non-Fermi liquids have been extensively investigated, both from experimental and theoretical point of view. In particular, one of the most spectacular feature appearing in non-Fermi liquids is the phenomenon of Spin-Charge (SC) separation, which is surely relevant for the physics
It is well established that at low energies one-dimensional (1D) fermionic systems are described by the Luttinger-liquid (LL) theory, which predicts phenomena such as spin-charge separation and charge fractionalization into chiral modes. Here we show through the time evolution of an electron injected into a 1D ttt-JJJ model, obtained with time-dependent density matrix renormalization group, that a further fractionalization of both charge and spin takes place beyond the hydrodynamic limit. Its dynamics can be understood at the supersymmetric point (J=2tJ=2tJ=2t) in terms of the excitations of the Bethe-ansatz solution. Furthermore we show that fractionalization with similar characteristics extends to the whole region corresponding to a repulsive LL.
We apply field theoretical renormalization group (RG) methods to describe the Tomonaga–Luttinger model as an important test ground to deal with spin–charge separation effects in higher spatial dimensions. We calculate the RG equations for the renormalized forward couplings g 2 R and g 4 R up to two-loop order and demonstrate that they do not flow in the vicinities of the Fermi points (FPs). We solve the Callan–Symanzik equation for G ( a ) R in the vicinities of the FPs. We calculate the related spectral function and the momentum distribution function at p = k F , p 0 = ω. We compute the renormalized one-particle irreducible function Γ (2) (+) R ( p , p 0 = 0; Λ) and show that it carries important spin–charge separation effects in agreement with well-known results. Finally, we discuss the implementation of the RG scheme taking into account the important simplifications produced by the Ward identities.
The low-energy properties of interacting Fermi systems are highly constrained by conservation laws. They generally simplify the structure of the underlying renormalization group by reducing the number of independent renormalization constants. In one dimension, all properties of normal metallic fixed points are uniquely determined by separate charge and spin conservation for states near the left and right Fermi points, respectively. We construct the general Luttinger-liquid theory of one-dimensional (1D) metals directly from these conservation laws. Luttinger-liquid parameters emerge naturally from the velocities associated with the conserved currents at the Luttinger-liquid fixed point. Instead of bosonization, one may thus use techniques familiar from Fermi-liquid theory, i.e., Feynman diagrams, equations of motion, and Ward identities. The choice of a technique comprising both Fermi- and Luttinger-liquid theory makes the similarities and differences of both theories particularly transparent, and sets the stage for constructing non-Fermi-liquid metallic fixed points in d>1. Several generic properties and asymptotic conservation laws of 2D non-Fermi-liquid metals are discussed.
One of the key features of one-dimensional systems is the phenomenon of spin–charge separation. We show this separation of spin and charge in real time in the 1D Hubbard model by the application of the adaptive time-dependent density-matrix renormalization group method. It is found for single particle excitations and density wave packets far beyond the low-energy Luttinger liquid theory. An experimental set-up is proposed on how to observe the spin–charge separation with ultracold Fermi gases in a harmonic trap using the presence of the Mott-insulator region. Quantitative estimates for experimental parameters are given.
In the present work, we implement an explicit two-loop renormalization of a two-dimensional flat Fermi surface (FS) in the framework of a field theoretical renormalization group approach (RG). In our scheme, we derive the RG equations for both the coupling functions and Fermi energy. In this way, we are able to probe the existence of spin-charge separation by showing that the low-energy sector of the system is in fact a non-Fermi liquid. In addition, associating the true interacting FS to the infrared stable (IR) fixed point of the Fermi energy, we demonstrate here that it either acquires a small curvature and behaves as a "Luttinger liquid" or it suffers a truncation in k-space depicting an insulating spin liquid.
The authors introduce a new representation of the Bethe ansatz solutions for 1D Luttinger liquids which describes the spectral properties and asymptotic behaviour of the correlation functions of the 1D Hubbard model in terms of the renormalized interaction of charge and spin pseudoparticles. The study of the low-lying eigenstates is reduced to the familiar languages of band theory and of the Fermi liquid. The authors results provide a better understanding of the decoupling of charge and spin degrees of freedom in 1D interacting systems, generalize the concept of a Landau liquid and may be relevant to the physics of higher dimensional systems.