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arXiv:cond-mat/0310527 · Journal of Physics A Mathematical and General

Conformal field theory in the Tomonaga–Luttinger model with the 1/rβlong-range interaction

Hitoshi Inoue, K. Nomura

Published Feb 15, 2006• 16 citations
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Abstract

We attempt to construct U (1) conformal field theory (CFT) in the Tomonaga–Luttinger (TL) liquid with the 1/ r β long-range interaction (LRI). Treating the long-range forward scattering as a perturbation and applying CFT to it, we derive finite size scalings which depend on the power of the LRI. The obtained finite size scalings give nontrivial behaviours when β is odd and is close to 2. We find consistency between the analytical arguments and numerical results in the finite size scaling of energy.

DOI
10.1088/0305-4470/39/9/012

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We attempt to construct U (1) conformal field theory (CFT) in the Tomonaga–Luttinger (TL) liquid with the 1/ r β long-range interaction (LRI). Treating the long-range forward scattering as a perturbation and applying CFT to it, we derive finite size scalings which depend on the power of the LRI. The obtained finite size scalings give nontrivial behaviours when β is odd and is close to 2. We find consistency between the analytical arguments and numerical results in the finite size scaling of energy.
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We discuss effects of 1/rβ1/r^β1/rβ type long-range (LR) interactions in a tight-binding model by utilizing the bosonization technique, renormalization group and conformal field theory (CFT). We obtain the low energy action known for Kibble's model which generates the mass gap in 3 dimension when β=1β=1β=1, the Coulomb force case. In one dimension, the dispersion relations predict that the system remains gapless even for β=1β=1β=1 and the existences of Tomonaga-Luttinger liquid (TLL) when β>1β> 1β>1. When β=1β=1β=1, the LR interactions break TLL in the long wavelength limits, even if the strength is very small. We make the more precise arguments from the stand point of the renormalization group and CFT. Finally we derive the accurate finite size scaling of energies and thermodynamics. Moreover we proceed to numerical calculations, considering the LR umklapp process terms. We conclude that the TLL phase become wider in the strength space of interactions as the power βββ approaches to 1.
We demonstrate the universal properties of dissipative Tomonaga-Luttinger (TL) liquids by calculating correlation functions and performing finite-size scaling analysis of a non-Hermitian XXZ spin chain as a prototypical model in one-dimensional open quantum many-body systems. Our analytic calculation is based on effective field theory with bosonization, finite-size scaling approach in conformal field theory, and the Bethe-ansatz solution. Our numerical analysis is based on the density-matrix renormalization group generalized to non-Hermitian systems (NH-DMRG). We uncover that the model in the massless regime with weak dissipation belongs to the universality class characterized by the complex-valued TL parameter, which is related to the complex generalization of the c=1c=1c=1 conformal field theory. As the dissipation strength increases, the values of the TL parameter obtained by the NH-DMRG begin to deviate from those obtained by the Bethe-ansatz analysis, indicating that the model becomes massive for strong dissipation. Our results can be tested with the two-component Bose-Hubbard system of ultracold atoms subject to two-body loss.
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.
Finite-size scaling analysis is performed for one-dimensional Bose and Fermi systems with long-range interaction g/r2{\mathit{r}}^{2}r2, based on the exact solution of Sutherland. The low-energy behavior of this model is shown to be described by the c=1 conformal field theory. Exact formulas for various correlation exponents are obtained. At a special point g=4, the model belongs to the same universality class as the Haldane-Shastry model, the effective theory of which is the level-1 SU(2) Kac-Moody theory.
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.