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arXiv:cond-mat/9505033 · Physical Review Letters

Bosonization of One-Dimensional Exclusons and Characterization of Luttinger Liquids

Yong-Shi Wu, Yue Yu

Published Jul 31, 1995• 57 citations
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Abstract

We achieve a bosonization of one-dimensional ideal gas of particles obeying exclusion statistics $\ensuremath{\lambda}$ (so called $\ensuremath{\lambda}$ exclusons) at low temperatures, resulting in a new variant of $c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ conformal field theory with compactified radius $R\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1/\ensuremath{\lambda}}$. These ideal excluson gases exactly reproduce the low- $T$ critical properties of Luttinger liquids, so they can be used to characterize the fixed points of the latter. Generalized ideal gases with mutual statistics and nonideal gases with Luttinger-type interactions have also similar low- $T$ behavior, controlled by an effective statistics varying in a fixed-point line.

DOI
10.1103/physrevlett.75.890

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We achieve a bosonization of one-dimensional ideal gas of particles obeying exclusion statistics \ensuremathλ\ensuremath{\lambda}\ensuremathλ (so called \ensuremathλ\ensuremath{\lambda}\ensuremathλ exclusons) at low temperatures, resulting in a new variant of c=1c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1c=1 conformal field theory with compactified radius R=1/\ensuremathλR\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1/\ensuremath{\lambda}}R=1/\ensuremathλ​. These ideal excluson gases exactly reproduce the low- TTT critical properties of Luttinger liquids, so they can be used to characterize the fixed points of the latter. Generalized ideal gases with mutual statistics and nonideal gases with Luttinger-type interactions have also similar low- TTT behavior, controlled by an effective statistics varying in a fixed-point line.
We consider the concept of fractons as particles or quasiparticles which obey a specific fractal statistics in connection with a one-dimensional Luttinger liquid theory. We obtain a dual statistics parameter = ν + 1 which is identified with the controlling parameter e -2φ of the Luttinger model. In this way, a bosonic system characterized by a fractal index i f [ h ] = i f [2] = 1 is considered in a conformal field theory with central charge c [ν = 0] = 1 = i f [2] with a compactified radius R = 1/ 1/2 = 1. Thus, we have a mapping of a bosonic theory to a fermionic one and vice versa, i.e. the duality symmetry = 3- h of the universal class h of fractons defined in the interval 1< h <2 is satisfied.
The bosonization technique is useful for solving the interacting problem and for extracting a certain number of properties such as power law behavior of the correlation functions. However, this derivation is, strictly speaking, based on a linearization of the spectrum close to the Fermi energy. One can thus wonder what will remain of the results of the previous chapter when it is impossible to stay confined close to the Fermi level, for example, when the interactions become strong. In order to answer this question, it is necessary to rederive the bosonization formulas, in a more phenomenological but more general way. This chapter demonstrates that a similar concept as Fermi liquids exists in one dimension. This concept, called Luttinger liquid, encompasses most of the physics of simple one-dimensional systems and will be the starting point for the study more complex situations. The concept of Luttinger liquids is explained using very low level techniques.
We study in this paper the properties of a many body system of fermions obeying exclusion-statistics (Haldane liquid) where the origin of exclusion statistics is coming from an interaction-induced displacement field ak\mathbf{a}_{\mathbf{k}}ak​ introduced by Sutherland (Haldane-Sutherland liquid). In particular we show how the Luttinger Theorem becomes compatible with exclusion statistics as a result of momentum conservation and adiabaticity. As a result, the low energy properties of Haldane-Sutherland liquids are Fermi/Luttinger liquid-like.
We present a study of a gas of bosons confined in one dimension with Feshbach-resonant interactions, at zero temperature. Unlike the gas of one-dimensional bosons with non resonant interactions, which is known to be equivalent to a system of interacting spinless fermions and can be described using the Luttinger liquid formalism, the resonant gas possesses additional features. Depending on its parameters, the gas can be in one of three possible regimes. In the simplest of those, it can still be described by the Luttinger liquid theory, but its Fermi momentum cannot be larger than a certain cutoff momentum dependent on the details of the interactions. In the other two regimes, it is equivalent to a Luttinger liquid at low density only. At higher densities its excitation spectrum develops a minimum, similar to the roton minimum in helium, at momenta where the excitations are in resonance with the Fermi sea. As the density of the gas is increased further, the minimum dips below the Fermi energy, thus making the ground state unstable. At this point the standard ground state gets replaced by a more complicated one, where not only the states with momentum below the Fermi points, but also the ones with momentum close to that minimum, get filled, and the excitation spectrum develops several branches. We are unable so far to study this regime in detail due to the lack of the appropriate formalism.
Interactions are known to have dramatic effects on bosonic gases in one dimension (1D). Not only does the ground state transform from a condensate like state to an effective Fermi sea, but new fundamental excitations, which do not have any higher-dimensional equivalents, are predicted to appear. In this work, we trace these elusive excitations via their effects on the dynamical structure factor of 1D strongly interacting Bose gases at low temperature. An array of 1D Bose gases is obtained by loading a 87Rb^{87}\mathrm{Rb}87Rb condensate in a two-dimensional lattice potential. The dynamical structure factor of the system is probed by energy deposition through low-momentum Bragg excitations. The experimental signals are compared to recent theoretical predictions for the dynamical structure factor of the Lieb-Liniger model at T>0T>0T>0. Our results demonstrate that the main contribution to the spectral widths stems from the dynamics of the interaction-induced excitations in the gas, which cannot be described by the Luttinger liquid theory.