Luttinger Liquid

We review the theory of interacting Fermi systems whose low-energy physics is dominated by forward scattering, i.e. scattering processes generated by effective interactions with small momentum transfers. These systems include Fermi liquids as well as several important non-Fermi liquid phases: one-dimensional Luttinger liquids, systems with long-range interactions, and fermions coupled to a gauge field. We report results for the critical dimensions separating different "universality classes", and discuss the behavior of physical quantities such as the momentum distribution function, the single-particle propagator and low-energy response functions in each class. The renormalization group for Fermi systems will be reviewed and applied as a link between microscopic models and effective low-energy theories. Particular attention is payed to conservation laws, which constrain any effective low-energy theory of interacting Fermi systems. In scattering processes with small momentum transfers the velocity of each scattering particle is (almost) conserved. This asymptotic conservation law leads to non-trivial cancellations of Feynman diagrams and other simplifications, making thus possible a non-perturbative treatment of forward scattering via Ward identities or bosonization techniques.

A series of c-axis oriented YBa2Cu3Ox-films with different oxygen content were prepared by laser deposition. The oxygen content x was determined by X-ray diffraction and by resonant Rutherford-back-scattering (RRBS) measurements. The c-axis length in these films of YBa2Cu30 x is about 0.5% larger compared to bulk values. We describe transport measurements in magnetic fields up to 7 Tesla between room temperature and T~ in samples with an oxygen content between the orthorhombic-to-tetragonal transition (x ~ 6.4) and full oxygenation (x ~ 7). The ratio p/R R was investigated with respect to the "two-dimensional Luttinger liquid" theory and the model of the "two-dimensional ionic metal". We report on deviations from the expected quadratic temperature behaviour of p/RH, especially in films with high oxygen content.

The one-dimensional Holstein model of spinless fermions interacting with dispersionless phonons is studied using a new variant of the density matrix renormalization group. By examining various lowenergy excitations of finite chains, the metal-insulator phase boundary is determined precisely and agrees with the predictions of strong coupling theory in the anti-adiabatic regime and is consistent with renormalization group arguments in the adiabatic regime. The Luttinger liquid parameters, determined by finite-size scaling, are consistent with a Kosterlitz-Thouless transition.

We develop two theoretical approaches for dealing with the low-energy effects of the repulsive interaction in one-dimensional electron systems. Renormalization Group methods allow us to study the low-energy behavior of the unscreened interaction between currents of well-defined chirality in a strictly one-dimensional electron system. A dimensional regularization approach is useful, when dealing with the low-energy effects of the long-range Coulomb interaction. This method allows us to avoid the infrared singularities arising from the long-range Coulomb interaction at D = 1. We can also compare these approaches with the Luttinger model, in order to analyze the effects of the short range term in the interaction.

This article reviews the electronic and transport properties of carbon nanotubes. The focus is mainly theoretical, but when appropriate, the relation with experimental results is mentioned. While simple band-folding arguments will be invoked to rationalize how the metallic or semiconducting character of nanotubes is inferred from their topological structure, more sophisticated tight-binding and ab initio treatments will be introduced to discuss more subtle physical effects, such as those induced by curvature, tube-tube interactions or topological defects. The same approach will be followed for transport properties. The fundamental aspects of conduction regimes and transport length scales will be first briefly presented using simple models of disorder, with the derivation of a few analytic results concerning specific situations of short and long range static perturbations. Further, the latest developments in semi-empirical or ab initio simulations aiming at exploring the effect of realistic static scatterers (chemical impurities, adsorbed molecules, etc.) or inelastic electron-phonon interactions, will be emphasized. Finally, specific issues, going beyond the noninteracting electron model, will be addressed, including excitonic effects in optical experiments, the Coulomb blockade regime, and the Luttinger liquid, charge density waves or superconducting transitions.

The dynamical properties of hole motion in an antiferromagnetic background are determined in one dimensional models in zero magnetic field, where spin isotropy holds, as well as in an external magnetic field. The latter case is also relevant, via particle-hole transformation, to the problem of hole propagation in one dimensional "superconductors". The singularities in the spectral function are investigated by means of bosonization techniques and perturbation theories. Results are then compared with Bethe ansatz solutions and Lanczos diagonalizations. The formalism also leads to interesting connections to the single impurity problem in Luttinger liquids. A rich structure is found in the spectral function whenever spin isotropy is broken, suggesting the presence of exotic momentum dependence in photoemission spectra of (quasi) one dimensional materials.

We study an exactly solvable model describing a stripe consisting of a Toda array of N anharmonic elastic chains sandwiched between two conducting chains. It is shown that the presence of a charge on one chain generates a gapless excitation branch (Luttinger liquid) on the other and leads to increase in the phonon frequencies.

We investigate the rectification of an ac bias in Luttinger liquids in the presence of an asymmetric potential (the ratchet effect). We show that strong repulsive electron interaction enhances the ratchet current in comparison with Fermi liquid systems, and the dc I − V curve is strongly asymmetric in the low-voltage regime even for a weak asymmetric potential. At higher voltages the ratchet current exhibits an oscillatory voltage dependence.

We achieve a bosonization of one-dimensional ideal gas of exclusion statistics λ at low temperatures, resulting in a new variant of c = 1 conformal field theory with compactified radius R = 1/λ. 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 non-ideal gases with Luttinger-type interactions have also similar behavior, controlled by an effective statistics varying in a fixed-point line.

We study the effect of external potential on transport properties of the fermionic two-leg ladder model. The response of the system to a local perturbation is strongly dependent on the ground state properties of the system and especially on the dominant correlations. We categorise all phases and transitions in the model (for incommensurate filling) and introduce "hopping-driven transitions" that the system undergoes as the inter-chain hopping is increased from zero. We also describe the response of the system to an ionic potential. The physics of this effect is similar to that of the single impurity, except that the ionic potential can affect the bulk properties of the system and in particular induce true long range order.

The low temperature phase diagram of 1D weakly disordered quantum systems like charge or spin density waves and Luttinger liquids is studied by a full finite temperature renormalization group (RG) calculation. For vanishing quantum fluctuations this approach is amended by an exact solution in the case of strong disorder and by a mapping onto the Burgers equation with noise in the case of weak disorder, respectively. At zero temperature we reproduce the quantum phase transition between a pinned (localized) and an unpinned (delocalized) phase for weak and strong quantum fluctuations, respectively, as found previously by Fukuyama or Giamarchi and Schulz.

We study the tunneling density of states (TDOS) for a junction of three Tomonaga-Luttinger liquid wires. We show that there are fixed points which allow for the enhancement of the TDOS, which is unusual for Luttinger liquids. The distance from the junction over which this enhancement occurs is of the order of x = v/(2ω), where v is the plasmon velocity and ω is the bias frequency. Beyond this distance, the TDOS crosses over to the standard bulk value independent of the fixed point describing the junction. This finite range of distances opens up the possibility of experimentally probing the enhancement in each wire individually.

We measure and compare the electronic transport properties of individual multiwall carbon nanotubes ͑MWNTs͒ and individual single-wall carbon nanotubes ͑SWNTs͒, and SWNT networks of varying thickness. The thinnest SWNT networks, like the individual semiconducting SWNTs, show nonlinear current-voltage ͑I-V͒ characteristics at low temperatures with a current that can be modulated by a gate-source voltage. The overall temperature dependence of conductance in the transparent networks changes systematically as the thickness of the network increases and is consistent with hopping conduction. On the other hand, the thickest SWNT networks ͑freestanding film͒ show more metallic behavior: their I-V characteristics are linear with no gate-voltage effect, and a large fraction of their conductivity is retained at very low temperatures, consistent with tunneling through thin barriers separating metallic regions. We make a comparison with individual MWNTs, which in some cases show even greater retention of conductance at very low temperatures, but ͑unlike the thickest SWNT networks͒ no changeover to metallic temperature dependence at higher temperatures. The temperature dependence of conductance in individual MWNTs is consistent with a model involving conduction in the two outer shells.

We use bosonization methods to calculate the exact finite-temperature single-electron Green's function of a spinful Luttinger liquid confined by open boundaries. The corresponding local spectral density is constructed and analyzed in detail. The interplay between boundary, finite-size and thermal effects are shown to dramatically influence the low-energy properties of the system. In particular, the well-known zero-temperature critical behavior in the bulk always crosses over to a boundary dominated regime in the vicinity of the Fermi level. Thermal fluctuations cause an enhanced depletion of spectral weight for small energies ω, with the spectral density scaling as ω 2 for ω much less than the temperature. Consequences for photoemission experiments are discussed. 71.10.Pm, 71.27.+a

In the present work we aim to characterize the lattice configurations and the magnetic behavior in the incommensurate phase of spin-Peierls systems. This phase emerges when the magnetic exchange interaction is coupled to the distortions of an underlying triangular lattice and has its experimental realization in the quasi-one dimensional compound family TiOX (X = Cl, Br). With a simple model of spin-1/2 chains inserted in a planar triangular geometry which couples them elastically, we are able to obtain the uniform-incommensurate and incommensurate-dimerized phase transitions seen in these compounds. Moreover, we follow the evolution of the wave-vector of the distortions with temperature inside the incommensurate phase. Finally, we predict gapless spin excitations for the intermediate phase of TiOX compounds along with incommensurate spin-spin correlations, offering to experimentalists a candidate to observe an exotic Luttinger liquid-like behavior.

We present an effective field theory formulation for a class of condensed matter systems with crystalline structures for which some of the discrete symmetries of the underlying crystal survive the long distance limit, up to mesoscopic scales, and argue that this class includes interesting materials, such as Sidoped GaAs. The surviving symmetries determine a limited set of possible effective interactions, that we analyze in detail for the case of Si-doped GaAs materials. These coincide with the ones proposed in the literature to describe the spin relaxation times for the Si-doped GaAs materials, obtained here as a consequence of the choice of effective fields and their symmetries. The resulting low-energy effective theory is described in terms of three (six chiral) one-dimensional Luttinger liquid systems and their corresponding intervalley transitions. We also discuss the Mott transition within the context of the effective theory.

We review the properties of quasi-one-dimensional organic superconductors: the Bechgaard salts and their sulfur analogs in their normal phase precursor to long-range order. We go through the main observations made in the normal state of these systems at low magnetic field and tackle the issue of their description under the angles of the Fermi and Luttinger liquid pictures.

We calculate the nonequilibrium dynamic evolution of a one-dimensional system of two-component fermionic atoms after a strong local quench by using a time-dependent spin-density-functional theory. The interaction quench is also considered to see its influence on the spin-charge separation. It is shown that the charge velocity is larger than the spin velocity for the system of on-site repulsive interaction (Luttinger liquid), and vise versa for the system of on-site attractive interaction (Luther-Emery liquid). We find that both the interaction quench and polarization suppress the spin-charge separation.

We propose that the finite size of the Kondo screening cloud, ξK, can be probed by measuring the charge quantization in a one-dimensional system coupled to a small quantum dot. When the chemical potential, µ in the system is varied at zero temperature, one should observe charge steps whose locations are at values of µ that are controlled by the Kondo effect when the system size L is comparable to ξK . We show that, if the standard Kondo model is used, the ratio between the widths of the Coulomb blockade valleys with odd or even number of electrons is a universal scaling function of ξK/L. If we take into account electron-electron interactions in a single-channel wire, this ratio also depends on the parameters of the effective Luttinger model; in addition, the scaling is weakly violated by a marginal bulk interaction. For the geometry of a quantum dot embedded in a ring, we show that the dependence of the charge steps on a magnetic flux through the ring is controlled by the size of the Kondo screening cloud.

The influence of randomly distributed point impurities and planar defects on order and transport in type-II superconductors and related systems is studied. It is shown that the Bragg glass phase is unstable with respect to planar defects. Even a single weak defect plane oriented parallel to the magnetic field as well as to one of the main axis of the Abrikosov flux line lattice is a relevant perturbation in the Bragg glass. A defect that is aligned with the magnetic field restores the flux density oscillations which decay algebraically with the distance from the defect. The theory exhibits striking similarities to the physics of a Luttinger liquid with a frozen impurity. The exponent for the flux line creep in the direction perpendicular to a relevant defect is derived. We find that the flux line lattice exhibits in the presence of many randomly distributed parallel planar defects aligned to the magnetic field a new glassy phase which we call planar glass. The planar glass is characterized by diverging shear and tilt moduli, a transverse Meissner effect, resistance against shear deformations. We also obtain sample to sample fluctuations of the longitudinal magnetic susceptibility and an exponential decay of translational long range order in the direction perpendicular to the defects. The flux creep perpendicular to the defects leads to a nonlinear resistivity ρ(J → 0) ∼ exp[−(JD/J) 3/2 ]. Strong planar defects enforce arrays of dislocations that are located at the defects with a Burgers vector parallel to the defects in order to relax shear strain.

We develop a theoretical framework that applies to the intermediate regime between the Coulomb blockade and the Luttinger liquid behavior in multi-walled carbon nanotubes [1]. We show that, in the crossover regime, the tunneling conductance follows a power-law behavior as a function of the temperature, with an exponent that oscillates with the gate voltage as observed in the experiments. We also evaluate the effects of a transverse magnetic field on the transport properties of the multi-walled nanotubes. For fields of the order of 4 T, we already find important changes in the band structure of the outer shells. We then predict the appearance of sensible modulations in the exponent of the conductance for higher magnetic fields, as the different subbands are shifted towards the formation of flat Landau levels. [1] S. Bellucci, J. Gonzalez and P. Onorato, Phys. Rev. Lett. 95, 186403 (2005).

An model of La2−xSrxCuO4 explaining the features of incommensurate spin textures without any assumption of stripe formation is proposed. The foundations of this model are the mechanism of negative-U center formation proposed earlier and the concept of specific ordering of doped ions. It is shown that within the framework of the proposed model the features of "stripe" textures of La2−xSrxCuO4 reflect exclusively the geometrical relations existing in a square lattice and the competition between different types of doped hole ordering.

We determine the phase diagram and the momentum distribution for a one-dimensional Bose gas with repulsive short range interactions in the presence of a two-color lattice potential, with incommensurate ratio among the respective wave lengths, by using a combined numerical (DMRG) and analytical (bosonization) analysis. The system displays a delocalized (superfluid) phase at small values of the intensity of the secondary lattice V2 and a localized (Bose glass-like) phase at larger intensity V2. We analyze the localization transition as a function of the height V2 beyond the known limits of free and hard-core bosons. We find that weak repulsive interactions unfavor the localized phase i. e. they increase the critical value of V2 at which localization occurs. In the case of integer filling of the primary lattice, the phase diagram at fixed density displays, in addition to a transition from a superfluid to a Bose glass phase, a transition to a Mott-insulating state for not too large V2 and large repulsion. We also analyze the emergence of a Bose-glass phase by looking at the evolution of the Mott-insulator lobes when increasing V2. The Mott lobes shrink and disappear above a critical value of V2. Finally, we characterize the superfluid phase by the momentum distribution, and show that it displays a power-law decay at small momenta typical of Luttinger liquids, with an exponent depending on the combined effect of the interactions and of the secondary lattice. In addition, we observe two side peaks which are due to the diffraction of the Bose gas by the second lattice. This latter feature could be observed in current experiments as characteristics of pseudo-random Bose systems.

In this paper I discuss a mechanism for ferrimagnetism in (1+1)-dimensions. The mechanism is related to a special class of interactions described by operators with non-zero Lorentz spin. Such operators are present in such problems as the problem of tunneling between Luttinger liquids and the problem of frustrated spin ladder. Exact solutions are presented for a representative class of models possessing a continuous isotopic symmetry. It is shown that the interactions (i) dynamically generate static oscillations with the wave vector dependent on the coupling constant, (ii) give rise to spontaneous breaking of this symmetry at T = 0 accompanied by generation of the magnetic moment and appearence of gapless modes with a non-relativistic (ferromagnetic) dispersion E ∼ k 2 , (iii) generate massive (roton) modes.

This paper is written as a brief introduction for beginning graduate students. The picture of electron waves moving in a cristalline potential and interacting weakly with each other and with cristalline vibrations suffices to explain the properties of technologically important materials such as semiconductors and also simple metals that become superconductors. In magnetic materials, the relevant picture is that of electrons that are completely localized, spin being left as the only relevant degree of freedom. A number of recently discovered materials with unusual properties do not fit in any of these two limiting cases. These challenging materials are generally very anisotropic, either quasi one-dimensional or quasi two-dimensional, and in addition their electrons interact strongly but not enough to be completely localized. High temperature superconductors and certain organic conductors fall in the latter category. This paper discusses how the effect of low dimension leads to new paradigms in the one-dimensional case (Luttinger liquids, spin-charge separation), and indicates some of the attempts that are being undertaken to develop, concurrently, new methodology and new concepts for the quasi-two-dimensional case, especially relevant to high-temperature superconductors.

We reconsider the theory of dc and ac interchain conductivity in quasi-one dimensional systems. Our results are in good agreement with the measured c-axis optical conductivity of (TMTSF)2ClO4 and suggest that the c-axis dc-conductivity of (TMTSF)2PF6 in the 150K < T < 300K range is dominated by precursor effects of Mott localization. The crossover from a Luttinger liquid at high energy to a Fermi liquid at low energy is also addressed, within a dynamical mean-field theory. Implications for the inter-chain resistivity and Drude weight in the Fermi liquid regime are discussed.

This study builds upon the work of Palacios and MacDonald (Phys. Rev. Lett. 76, 118 (1996)), wherein they identify the bosonic excitations of Wen's approach for the edge of the 1/3 fractional quantum Hall state with certain operators introduced by Stone. Using a quantum Monte Carlo method, we extend to larger systems containing up to 40 electrons and obtain more accurate thermodynamic limits for various matrix elements for a short range interaction. The results are in agreement with those of Palacios and MacDonald for small systems, but offer further insight into the detailed approach to the thermodynamic limit. For the short range interaction, the results are consistent with the chiral Luttinger liquid predictions. We also study excitations using the Coulomb ground state for up to nine electrons to ascertain the effect of interactions on the results; in this case our tests of the chiral Luttinger liquid approach are inconclusive.

We continue the study of the entanglement entropy of two disjoint intervals in conformal field theories that we started in J. Stat. Mech. (2009) P11001. We compute Trρ n A for any integer n for the Ising universality class and the final result is expressed as a sum of Riemann-Siegel theta functions. These predictions are checked against existing numerical data. We provide a systematic method that gives the full asymptotic expansion of the scaling function for small fourpoint ratio (i.e. short intervals). These formulas are compared with the direct expansion of the full results for free compactified boson and Ising model. We finally provide the analytic continuation of the first term in this expansion in a completely analytic form.

It is shown that the Bragg glass phase can become unstable with respect to planar defects. A single defect plane that is oriented parallel to the magnetic field as well as to one of the main axis of the Abrikosov flux line lattice is always relevant, whereas we argue that a plane with higher Miller index is irrelevant, even at large defect potentials. A finite density of parallel defects with random separations can be relevant even for larger Miller indices. Defects that are aligned with the applied field restore locally the flux density oscillations which decay algebraically with distance from the defect. The current voltage relation is changed to ln V (J) ∼ −J −1 . The theory exhibits some similarities to the physics of Luttinger liquids with impurities. PACS numbers: 74.25.Qt, 61.72.Mm, 72.15.Rn Type-II superconductors have to contain a certain amount of disorder to sustain superconductivity: the disorder pins magnetic flux lines, hence preventing dissipation due to their motion . For some time it was believed that disorder due to randomly distributed impurities destroys the long range translational order (LRO) of the Abrikosov flux line lattice [3]. More recently it was shown that the effect of impurities is much weaker resulting in a phase with quasi-LRO, the so-called Bragg glass . In this phase the averaged flux line density is constant but a remnant of its periodic order is seen in the correlations of the oscillating part of the density which decay as a power law. Experimental signatures of this phase have been observed . An important feature of the Bragg glass is the highly non-linear current-voltage relation related to the flux creep which is of the form ln V (J) ∼ −J −1/2 so that the linear resistance vanishes.

Presents a technique to directly excite Luttinger liquid collective modes in carbon nanotubes at gigahertz frequencies. By modeling the nanotube as a nano-transmission line with distributed kinetic and magnetic inductance as well as distributed quantum and electrostatic capacitance, we calculate the complex frequency-dependent impedance for a variety of measurement geometries. Exciting voltage waves on the nano-transmission line is equivalent to directly exciting the yet-to-be observed one-dimensional plasmons, the low energy excitation of a Luttinger liquid. Our technique has already been applied to two-dimensional plasmons and should work well for one-dimensional plasmons. Tubes of length 100 microns must be grown for gigahertz resonance frequencies. Ohmic contact is not necessary with our technique; capacitive contacts can work. Our modeling has applications in potentially terahertz nanotube transistors and RF nanospintronics.

We investigate the electronic instabilities in carbon nanotubes (CNs), looking for the break-down of the one dimensional Luttinger liquid regime due to the strong screening of the long-range part of the Coulomb repulsion. We show that such a breakdown is realized both in ultra-small single wall CNs and multi wall CNs, while a purely electronic mechanism could explain the superconductivity (SC) observed recently in ultra-small (diameter $ \sim 0.4 nm$) single wall CNs ($T_c\sim 15 ^{o}K$) and entirely end-bonded multi-walled ones ($T_c\sim 12 ^{o}K$). We show that both the doping and the screening of long-range part of the electron-electron repulsion, needed to allow the SC phase, are related to the intrinsically 3D nature of the environment where the CNs operate.

We show how the main features of the electronic transport properties of single-wall carbon nanotube (SWNT) networks can be understood in terms of a simple model involving metallic conduction interrupted by thin tunnelling barriers, backscattering by zoneboundary phonons, and variable range hopping. Within this framework we examine the effect of reducing the thickness of the SWNT networks, chemical treatments, and ion irradiation. The conduction mechanism can be tuned from variable-range hopping between localized states (for the thinnest networks), to metallic conduction interrupted by thin barriers through which conduction is by tunnelling (for thick freestanding films). Chemical treatment of the thick films by different molecules leads to retention of metallic character but changes (increases or decreases) the charge carrier density. For ion-irradiated thick films we find two competing effects: a change to hopping-type conduction in the direct impact layer that lowers conductivity, and annealing effects extending deeper than the ion penetration depth that increase conductivity and lead to a peak in conductivity as a function of irradiation dose. We briefly discuss current-voltage characteristics and possible Luttinger liquid effects. r

In a variety of semimetaIs with vanishing density of s~ates a the Fermi level, the low energy electronic states are described by an effective Dirac equation, instead of the more conventional effective mass appromimation (equivalent to SchrJdinger equation}. When this happens, electron-electron ir~teractions give vise ~o anomalous features, which differ from those of conventional metals and insulators. A comprehensive discussion of these effects is presented, making use of the fact that Dirac electrons ir~teracting through Coulomb forces lead to a well defined, renormalizable, quantum field theory, which has many similarities to quantum electrodynamics. The most remarkable feature is the absence of a coherent quasipartiele pole, like in 1D Luttinger liquids.

molecular transistors and quantum wires formed in two-dimensional electron gas. The review starts with a textbook description of resonant tunneling of noninteracting electrons through a double-barrier structure. The effects of electron–electron interaction in sequential and resonant electron tunneling are studied by using Luttinger liquid model of electron transport in quantum wires. The experimental aspects of the problem (fabrication of quantum wires and transport measurements) are also considered. The influence of vibrational and electromechanical effects on resonant electron tunneling in molecular transistors is discussed. PACS: 73.63.–b Electronic transport in nanoscale materials and structures; 73.23.Hk Coulomb blockade; single-electron tunneling; 85.85.+j Micro- and nano-electromechanical systems (MEMS/NEMS) and devices.

Neutron diffraction is used to investigate the field-induced, antiferromagnetically ordered state in the two-leg spin-ladder material (Hpip)2CuBr4. This "classical" phase, a consequence of weak interladder coupling, is nevertheless highly unconventional: its properties are influenced strongly by the spin Luttinger-liquid state of the ladder subunits. We determine directly the order parameter (transverse magnetization), the ordering temperature, the spin structure, and the critical exponents around the transition. We introduce a minimal, microscopic model for the interladder coupling and calculate the quantum fluctuation corrections to the mean-field interaction.

The low-energy effective Hamiltonian of three coupled spin chains with periodic boundary conditions (spin tube) is expressed, in the limit of strong interchain coupling, in terms of XXZ chains coupled by biquadratic exchange interaction. A similar effective model was also proposed to describe the coupling of spins to orbital degrees of freedom in materials such as NaV2O5. We investigate the effective model by means of bosonization and renormalization group techniques, and find that the generic phase diagram comprises a gapless region and gapped regions consisting of a spin liquid phase and various antiferromagnetic phases. We discuss the properties of the spin liquid phase, in particular the nature of the ground state and of the elementary excitations above it. We then study the effect of a magnetic field, and conclude that a strong enough magnetic field can suppress the dimerized phase leading to a two component Luttinger liquid. The critical exponents at the transition gapful-gapless are calculated and shown to be non-universal in the spin tube case or the generic spin orbital problem.

Carbon nanotubes (CNTs) provide an ideal medium for testing the behavior of one-dimensional electron systems and are promising candidates for electronic applications such as sensors or field-effect transistors. This thesis describes the use of low frequency resistance fluctuations to measure both the properties of the onedimensional electron system in CNTs, and the sensitivity of CNT devices to their environment. Low frequency noise was measured in CNTs in field effect transistor (FET) geometry. CNTs have a large amount of surface area relative to their volume and are expected to be strongly affected by their environment, leading to speculation that CNTs should have large amounts of 1/f noise. My measurements indicate that the noise level is in the same range as that of traditional FETs, an encouraging result for possible electronic applications. The temperature dependence of 1/f noise from 1.2 K to 300 K can be used to extract the characteristic energies of the fluctuators

We present a 14 N nuclear magnetic resonance study of a single crystal of CuBr4(C5H12N)2 (BPCB) consisting of weakly coupled spin-1/2 Heisenberg antiferromagnetic ladders. Treating ladders in the gapless phase as Luttinger liquids, we are able to fully account for (i) the magnetic field dependence of the nuclear spin-lattice relaxation rate T −1 1 at 250 mK and for (ii) the phase transition to a 3D ordered phase occuring below 110 mK due to weak interladder exchange coupling. BPCB is thus an excellent model system where the possibility to control Luttinger liquid parameters in a continuous manner is demonstrated and Luttinger liquid model tested in detail over the whole fermion band.

The conductivity properties between Luttinger liquids are analyzed by exact Renormalization Group methods. We prove that in a two chain system or in a model of bilayer graphene, described by two coupled fermionic honeycomb lattices interacting with a gauge field, the transverse optical conductivity at finite temperature is anomalous and decreasing together with the frequency as a power law with Luttinger liquid exponent.