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International Journal of Theoretical and Mathematical Physics

Power-Law Dependence of Correlaration Functions in the Tomonaga-Luttinger Liquid

Nelson Oyindenyifa Nenuwe, Famous Akpojotor

Published Jan 1, 2015• 1 citation
OpenAlex

Abstract

The power-law dependence of correlation functions at low energies is what clearly distinguishes the Tomonaga-Luttinger (TL) liquid and the Fermi liquid (FL). The correlation function critical exponent for the FL is usually fixed to an integer, while that of the TL liquid varies continuously. We calculate the nature of power-law dependence in the charge, spin and electron correlation functions for the TL liquid at both zero and small magnetic fields. In particular, our approach centres on the Bethe Ansatz and conformal field theory (CFT). This is important since the concept of CFT relates critical exponents of various correlation functions with the finite-size corrections to the energy spectrum. Our findings show that increase in the particle density nc, and external magnetic field B, increases the value of the critical exponents while increase in the interaction strength u, decreases the value of the critical exponents.

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The power-law dependence of correlation functions at low energies is what clearly distinguishes the Tomonaga-Luttinger (TL) liquid and the Fermi liquid (FL). The correlation function critical exponent for the FL is usually fixed to an integer, while that of the TL liquid varies continuously. We calculate the nature of power-law dependence in the charge, spin and electron correlation functions for the TL liquid at both zero and small magnetic fields. In particular, our approach centres on the Bethe Ansatz and conformal field theory (CFT). This is important since the concept of CFT relates critical exponents of various correlation functions with the finite-size corrections to the energy spectrum. Our findings show that increase in the particle density nc, and external magnetic field B, increases the value of the critical exponents while increase in the interaction strength u, decreases the value of the critical exponents.
Contrary to the usual Fermi liquids, where the exponent of the momentum distribution at k=kF is fixed to an integer, for the Tomonaga-Luttinger (TL) liquids the anomalous exponent of the momentum distribution changes continuously, resulting in a power-law singularity of the momentum distribution function. It has been said that this power-law singularity appears at kF and 3kF for the Hubbard model as well as at kF for the t-J model. Using the conformal field theory (CFT) technique, we present an exact calculation of the anomalous exponent of the t-J model at kF, 3kF, 5kF and 7kF and compare it with the results for the Hubbard model.
The correlation functions of one-dimensional Hubbard model in the presence of external magnetic field was investigated through the conformal field technique. The long distance behaviour of the correlation functions and their critical exponents for the model in the presence of a magnetic field are established by solving the dressed charge matrix equations and setting the number of occupancies to one, as an alternative to the usual zero often used by authors in literatures. Our result shows the critical exponents of the correlation function grows monotonically with magnetic field and reduces to definite values at zero magnetic field around various Fermi points.
The critical properties of Tomonaga-Luttinger liquids including a mobile impurity are studied by means of the Bethe ansatz method, conformal field theory (CFT), and bosonization. We calculate conformal dimensions by performing the CFT analysis of the energy spectrum for an integrable impurity model, and show that this system is classified as c=1c=1c=1 shifted Gaussian CFT. The exact critical exponents of various correlation functions are obtained as functions of the impurity mass and its momentum. Passing to the continuum limit, we then construct an effective field theory in terms of bosonization, and directly calculate some correlation functions. A possible application to the Fermi-edge singularity in quantum wires is also discussed.
The electronic system of the 1D Hubbard model is not stable due to Peierls instability; the correlations are strong even for the weak Coulomb interaction. The resulting strongly correlated state without Landau quisi--particle excitations is known as the Luttinger liquid. Critical exponents of a power--law dependence of correlation functions at low energies differ substantially for the Luttinger and Fermi liquids. In this paper we evaluate two critical exponents that define non--trivial behavior of the density of electronic states at low frequences and the momentum distribution of the occupation number nearby kFk_FkF​.
For the one-dimensional Holstein model, we show that the relations among the scaling exponents of various correlation functions of the Tomonaga Luttinger liquid (LL), while valid in the thermodynamic limit, are significantly modified by finite-size corrections. We obtain analytical expressions for these corrections and find that they decrease very slowly with increasing system size. The interpretation of numerical data on finite-size lattices in terms of LL theory must therefore take these corrections into account. As an important example, we re-examine the proposed metallic phase of the zero-temperature, half-filled one-dimensional Holstein model without employing the LL relations. In particular, using quantum Monte Carlo calculations, we study the competition between the singlet pairing and charge ordering. Our results do not support the existence of a dominant singlet pairing state.
Universal scaling relations for dissipative Tomonaga-Luttinger (TL) liquids with SU(N\mathrm{SU}(NSU(N) spin symmetry are obtained for both fermions and bosons, by using asymptotic Bethe-ansatz solutions and conformal field theory (CFT) in one-dimensional non-Hermitian quantum many-body systems with SU(N\mathrm{SU}(NSU(N) symmetry. We uncover that the spectrum of dissipative TL liquids with SU(N\mathrm{SU}(NSU(N) spin symmetry is described by the sum of one charge mode characterized by a complex generalization of c=1c=1c=1 U(1) Gaussian CFT, and $N\ensuremath{-}1$ spin modes characterized by level-1SU(N1\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N1SU(N) Kac-Moody algebra with the conformal anomaly $c=N\ensuremath{-}1$, and thereby dissipation only affects the charge mode as a result of spin-charge separation in one-dimensional non-Hermitian quantum systems. The derivation is based on a complex generalization of Haldane's ideal-gas description, which is implemented by the SU(N\mathrm{SU}(NSU(N) Calogero-Sutherland model with inverse-square long-range interactions.