Equilibration of Luttinger liquid and conductance of quantum wires

Luttinger liquid theory describes one-dimensional electron systems in terms of noninteracting bosonic excitations. In this approximation thermal excitations are decoupled from the current flowing through a quantum wire, and the conductance is quantized. We show that relaxation processes not captured by the Luttinger liquid theory lead to equilibration of the excitations with the current and give rise to a temperature-dependent correction to the conductance. In long wires, the magnitude of the correction is expressed in terms of the velocities of bosonic excitations. In shorter wires it is controlled by the relaxation rate.     

Effect of impurities on the heat capacity of quasi-one-dimensional conductors

The low-temperature heat capacity of a quasi-one-dimensional conductor in the Luttinger liquid state is studied for conductors with a charge density wave and without it. It is shown that pinning of the Luttinger liquid leads to localization of spins and a strong effect of the magnetic field on the thermodynamics of samples resembling the effect observed in recent experiments. We believe that this indicates the possibility of the formation of a Luttinger liquid in quasi-one-dimensional crystals.

     

One-dimensional quantum liquids with power-law interactions: the Luttinger staircase

We study one-dimensional fermionic and bosonic gases with repulsive power-law interactions 1/|x|(β), with β>1, in the framework of Tomonaga-Luttinger liquid (TLL) theory. We obtain an accurate analytical expression linking the TLL parameter to the microscopic Hamiltonian, for arbitrary β and strength of the interactions. In the presence of a small periodic potential, power-law interactions make the TLL unstable towards the formation of a cascade of lattice solids with fractional filling, a "Luttinger staircase." Several of these quantum phases and phase transitions are realized with ground state polar molecules and weakly bound magnetic Feshbach molecules.     

Equilibration of a one-dimensional quantum liquid

We review some of the recent results on equilibration of one-dimensional quantum liquids. The low-energy properties of these systems are described by the Luttinger liquid theory, in which the excitations are bosonic quasiparticles. At low temperatures, the relaxation of the gas of excitations toward full equilibrium is exponentially slow. In electronic Luttinger liquids, these relaxation processes involve backscattering of electrons and give rise to interesting corrections to the transport properties of one-dimensional conductors. We focus on the phenomenological theory of the equilibration of a quantum liquid and obtain an expression for the relaxation rate in terms of the excitation spectrum.     

Phase Diagram of the Spin-1/2 Heisenberg Alternating Chain in a Magnetic Field *

By using the infinite time-evolving block decimation, we study quantum fidelity and entanglemententropy in the spin-1/2 Heisenberg alternating chain under an external magnetic field .The effects of the magnetic field on the fidelity are investigated, and its relation with the quantum hase transition (QPT) is analyzed. The phase diagram of the model is given accordingly, which supports the Haldane phase, the singlet-dimer phase, the Luttinger liquid phase and the paramagnetic phase. The scaling of entanglement entropy in the gapless Luttinger liquid phase is studied, and the central charge $c=1$ is obtained. We also study the relationship between the quantum coherence, string order parameter and QPTs. Results obtained from these quantum information observations are consistent with the previous reports.     

Measuring Luttinger liquid correlations from charge fluctuations in a nanoscale structure

We suggest an experiment to study Luttinger liquid behavior in a one-dimensional nanostructure, avoiding the usual complications associated with transport measurements. The proposed setup consists of a quantum box, biased by a gate voltage, and side coupled to a quantum wire by a point contact. Close to the degeneracy points of the Coulomb blockaded box, and in the presence of a magnetic field sufficiently strong to spin polarize the electrons, the setup can be described as a Luttinger liquid interacting with an effective Kondo impurity. Using exact nonperturbative techniques, we predict that the differential capacitance of the box will exhibit distinctive Luttinger liquid scaling with temperature and gate voltage.     

Transport of charge-density waves in the presence of disorder: classical pinning versus quantum localization

We consider the interplay of the elastic pinning and the Anderson localization in the transport properties of a charge-density wave in one dimension, within the framework of the Luttinger model in the limit of strong repulsion. We address a conceptually important issue of which of the two disorder-induced phenomena limits the mobility more effectively. We argue that the interplay of the classical and quantum effects in transport of a very rigid charge-density wave is quite nontrivial: the quantum localization sets in at a temperature much smaller than the pinning temperature, whereas the quantum localization length is much smaller than the pinning length.     

有限温度下一维Gaudin-Yang模型的热力学性质

本文通过数值求解有限温度下一维均匀费米Gaudin-Yang模型的热力学Bethe-ansatz方程, 研究了此模型的基本性质,得到了在给定的温度或给定的相互作用下, 化学势、相互作用、粒子密度和熵的相互变化图像. 对结果分析发现, 在给定温度和相互作用下, 熵随着化学势的变化有一个量子临界区域.     

Luttinger-liquid behavior in tunneling through a quantum dot at zero magnetic field

Thermodynamic and transport properties of a two-dimensional circular quantum dot are studied theoretically at zero magnetic field. In the limit of a large confining potential, where the dot spectrum exhibits a shell structure, it is argued that both spectral and transport properties should exhibit Luttinger liquid behavior. These predictions are verified by direct numerical diagonalization. The experimental implications of such Luttinger liquid characteristics are discussed.     

Flow equations for the one-dimensional Kondo lattice model: static and dynamic ground state properties

The one-dimensional Kondo lattice model is investigated by means of Wegner's flow equation method. The renormalization procedure leads to an effective Hamiltonian which describes a free one-dimensional electron gas and a Heisenberg chain. The localised spins of the effective model are coupled by the well-known RKKY interaction. They are treated within a Schwinger boson mean field theory which permits the calculation of static and dynamic correlation functions. In the regime of small interaction strength static expectation values agree well with the expected Luttinger liquid behaviour. The parameter K_ρ of the Luttinger liquid theory is estimated and compared to recent results from density matrix renormalization group studies.     

Impurity-induced stabilization of Luttinger liquid in quasi-one-dimensional conductors

It is shown theoretically that the Luttinger liquid can exist in quasi-one-dimensional conductors in the presence of impurities in a form of a collection of bounded Luttinger liquids. The conclusion is based on the observation by Kane and Fisher that a local impurity potential in Luttinger liquid acts, at low energies, as an infinite barrier. This leads to a discrete spectrum of collective charge and spin density fluctuations, so that interchain hopping can be considered as a small parameter at temperatures below the minimum excitation energy of the collective modes. The results are compared with recent experimental observation of a Luttinger-liquid-like behavior in thin NbSe₃ and TaS₃ wires.     

The fractional quantum Hall effect: Chern-Simons mapping,duality, Luttinger liquids and the instanton vacuum

We derive, from first principles, the complete Luttinger liquid theory of abelian quantum Hall edge states. This theory includes disorder and Coulomb interactions as well as the coupling to external electromagnetic fields. We introduce a theory of spatially separated edge modes, find an enlarged dual symmetry and obtain a complete classification of quasiparticle operators and tunneling exponents. The chiral anomaly on the edge is used to obtain unambiguously the Hall conductance. In resolving the problem of counter-flowing edge modes, we find that the long range Coulomb interactions play a fundamental role. In order to set up a theory for arbitrary ν we use the idea of a two-dimensional network of percolating edge modes. We derive an effective, single mode Luttinger liquid theory for tunneling processes into the edge which yields a continuous tunneling exponent 1/ν. The network approach is also used to re-derive the instanton vacuum theory for plateau transitions.     

Signature of Mott-insulator transition with ultracold fermions in a one-dimensional optical lattice

Using the exact Bethe ansatz solution of the Hubbard model and Luttinger liquid theory, we investigate the density profiles and collective modes of one-dimensional ultracold fermions confined in an optical lattice with a harmonic trapping potential. We determine a generic phase diagram in terms of a characteristic filling factor and a dimensionless coupling constant. The collective oscillations of the atomic mass density, a technique that is commonly used in experiments, provide a signature of the quantum phase transition from the metallic phase to the Mott-insulator phase. A detailed experimental implementation is proposed.     

Renormalization of potential scattering in a weakly interacting one-dimensional electron gas

The conductance of a weakly interacting electron gas in the presence of a single scatterer is found at arbitrary strength of the scattering potential. At weak interaction one can use a simple renormalization group approach instead of the standard bosonization technique. For a model with spinless electrons this approach allows us to show explicitly the crossover from the Fermi-gas to the low-temperature Luttinger liquid behavior. Deviations from the Luttinger liquid theory are studied for a realistic model of spin- electrons.     

Quantum dissociation of an edge of a Luttinger liquid

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.     

Symanzik's method applied to fractional quantum Hall edge states

The method of separability, introduced by Symanzik, is applied in order to describe the effect of a boundary for a fractional quantum Hall liquid in the Laughlin series. An Abelian Chern‐Simons theory with plane boundary is considered and the Green functions both in the bulk and on the edge are constructed, following a rigorous, perturbative, quantum field theory treatment. We show that the conserved boundary currents find an explicit interpretation in terms of the continuity equation with the electron density satisfying the Tomonaga‐Luttinger commutation relation.     

Quantum fluctuation theorem in an interacting setup: point contacts in fractional quantum Hall edge state devices

We verify the validity of the Cohen-Gallavotti fluctuation theorem for the strongly correlated problem of charge transfer through an impurity in a chiral Luttinger liquid, which is realizable experimentally as a quantum point contact in a fractional quantum Hall edge state device. This is accomplished via the development of an analytical method to calculate the full counting statistics of the problem in all the parameter regimes involving the temperature, the Hall voltage, and the gate voltage.     

Hidden duality and associated instabilities of Tomonaga–Luttinger liquid on lattice

Hidden duality and its associated instabilities of the spinless Luttinger liquid on lattice are reported. The local quantum fluctuations due to the general multi-particle umklapp and other processes and the long-distance chiral modes compete and as a result produce a hierarchy of exotic charge density instabilities. Explicit bosonic quantum operators for the local density fluctuations are constructed and are used to make identification of the Luttinger liquid with the classical 2D Coulomb gas with -term and with the rich hidden duality.