In a Luttinger liquid phase of one-dimensional molecular matter the strength of zero-point motion can be characterized by dimensionless De Boer's number quantifying the interplay of quantum fluctuations and two-body interactions. Selecting the latter in the Morse form we show that dissociation of the Luttinger liquid is a process initiated at the system edge. The latter becomes unstable against quantum fluctuations at a value of De Boer's number which is smaller than that of the bulk instability which parallels the classical phenomenon of surface melting.
In a Luttinger liquid phase of one-dimensional molecular matter the strength of zero-point motion can be characterized by dimensionless De Boer's number quantifying the interplay of quantum fluctuations and two-body interactions. Selecting the latter in the Morse form we show that dissociation of the Luttinger liquid is a process initiated at the system edge. The latter becomes unstable against quantum fluctuations at a value of De Boer's number which is smaller than that of the bulk instability which parallels the classical phenomenon of surface melting.
Motivated by emerging experimental possibilities to confine atoms and molecules in quasi-one-dimensional geometries, we analyze ground-state properties of strictly one-dimensional molecular matter comprised of identical particles of mass m. Such a class of systems can be described by an additive two-body potential whose functional form is common to all substances which only differ in the energy $\ensuremath{\epsilon}$ and range l scales of the potential. With this choice De Boer's quantum theorem of corresponding states holds and the ground-state properties expressed in appropriate reduced form are only determined by the dimensionless parameter ${\ensuremath{\lambda}}_{0}^{2}\ensuremath{\sim}{\ensuremath{\Elzxh}}^{2}{/ml}^{2}\ensuremath{\epsilon},$ measuring the strength of zero-point motion in the system. The presence of a minimum in the two-body interaction potential leads to a many-body bound state which is a Luttinger liquid stable for not very large ${\ensuremath{\lambda}}_{0}.$ As ${\ensuremath{\lambda}}_{0}$ increases, the asymmetry of the two-body potential causes quantum expansion, softening, and eventual evaporation of the Luttinger liquid into a gas phase. Selecting the pair interaction potential in the Morse form we analytically compute the properties of the Luttinger liquid and its range of existence. We find that as ${\ensuremath{\lambda}}_{0}$ increases, the system first undergoes a discontinuous evaporation transition into a diatomic gas followed by a continuous dissociation transition into a monoatomic gas. In particular we find that spin-polarized isotopes of hydrogen and 3He are monoatomic gases, 4He is a diatomic gas, while molecular hydrogen and heavier substances are Luttinger liquids. We also investigate the effect of finite pressure on the properties of the liquid- and monoatomic gas phases. In particular we estimate a pressure at which molecular hydrogen undergoes an inverse Peierls transition into a metallic state which is a one-dimensional analog of the transition predicted by Wigner and Huntington in 1935 [E. Wigner and H.B. Huntington, J. Chem Phys. 3, 764 (1935)].
We describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.
We describe the relationship between quantum Hall edge states and the one-dimensional Luttinger liquid model. The Luttinger liquid model originated from studies of one-dimensional Fermi systems, however, it results that many ideas inspired by such a model can find applications to phenomena occurring even in higher dimensions. Quantum Hall systems which essentially are correlated two-dimensional electronic systems in a strong perpendicular magnetic field have an edge. It turns out that the quantum Hall edge states can be described by a one-dimensional Luttinger model. In this work, we give a general background of the quantum Hall and Luttinger liquid physics and then point out the relationship between the quantum Hall edge states and its one-dimensional Luttinger liquid representation. Such a description is very useful given that the Luttinger liquid model has the property that it can be bosonized and solved. The fact that we can introduce a simpler model of noninteracting bosons, even if the quantum Hall edge states of electrons are interacting, allows one to calculate exactly various quantities of interest. One such quantity is the correlation function which, in the asymptotic limit, is predicted to have a power law form. The Luttinger liquid model also suggests that such a power law exponent should have a universal value. A large number of experiments have found the quantum Hall edge states to show behavior consistent with a Luttinger liquid description. However, while a power law dependence of the correlation function has been observed, the experimental values of the exponent appear not to be universal. This discrepancy might be due to various correlation effects between electrons that sometimes are not easy to incorporate within a standard Luttinger liquid model.
A translation invariant one-dimensional system of spinless fermions with a finite-range attraction experiences a quantum phase transition to a phase-separated state. While being a conventional Luttinger liquid for a small interaction strength, spinless fermions form a droplet with the size smaller than the available one-dimensional volume when the interaction strength exceeds some critical value. A particularly remarkable signature of the droplet formation is the change in the lower edge of the many-body excitation spectrum. In the homogeneous phase, it has a Luttinger-liquid shape of periodic arcs on top of the shallow parabolic dispersion of the center of mass. When the interaction strength is increased, the arcs disappear completely as soon as the droplet is formed. We perform an exact diagonalization study of this system with the focus on the signatures of the quantum phase transition and the droplet properties. The one-particle and density-density correlation functions, the pressure, the sound velocity, and the droplet density are examined. The value of the critical interaction strength obtained from numerical data reasonably agrees with a simple mean-field analytical estimate. Due to the boson-fermion correspondence valid in one dimension, our results also hold for hard-core bosons with a finite-range attraction.
We study a system consisting of a Luttinger liquid coupled to a quantum dot on the boundary. The Luttinger liquid is expressed in terms of fermions interacting via density-density coupling, and the dot is modeled as an interacting resonant level onto which the bulk fermions can tunnel. We solve the Hamiltonian exactly and construct all eigenstates. We study both the zero- and finite-temperature properties of the system, in particular, we compute the exact dot occupation as a function of the dot energy in all parameter regimes. The system is seen to flow from weak to strong coupling for all values of the bulk interaction, with the flow characterized by a nonperturbative Kondo scale. We identify the critical exponents at the weak- and strong-coupling regimes.
We study a model of a quantum Hall system with the electrons confined to a narrow, linear channel. This system is mapped to a one-dimensional (1D) system which in the low-energy approximation has the form of a Luttinger liquid with different interactions between particles of equal and of opposite chiralities. The mapping is studied at the microscopic level, and we show in detail how the form of the 1D Luttinger liquid is detemined by interactions with particles at the edges and in the bulk of the quantum Hall system. We focus, in particular, on how the parameters are renormalized by the interactions between the two edges and show that the velocity parameter of the current is not modified. The paper is Paper I in a sequence of two papers, where Paper II [M. Horsdal and J. M. Leinaas, Phys. Rev. B 76, 195322 (2007)] gives a derivation of the two-dimensional electron correlation function.