Luttinger Liquid State Research Articles

Pairing of Cooper pairs in a fully frustrated Josephson-junction chain.

We study a one-dimensional Josephson-junction chain embedded in a magnetic field. We show that, when the magnetic flux per elementary loop equals half the superconducting flux quantum phi0 = h/2e, a local Z2 symmetry arises. This symmetry is responsible for a nematic Luttinger liquid state associated with bound states of Cooper pairs. We analyze the phase diagram and discuss some experimental possibilities to observe this exotic phase.

Electron Spin in Single Wall Carbon Nanotubes

We review aspects of electrical transport in metallic single wall carbon nanotubes (SWCNT) related to the spin of the conductance electrons. For large contact resistances, R ≫ h/2e2, a SWCNT exhibits Coulomb blockade, and transmission can only occur, when a gate voltage leads to an energy degeneracy for two different numbers of electrons in the SWCNT. The Coulomb blockade gate voltage change is directly proportional to the addition energy for single electron tunnelling. In certain ideal cases every second of the populated electronic states has a higher addition energy, indicating that two spindegenerate electrons are roomed at each orbital state. A low addition energy therefore corresponds to approaching an even number of electrons. The odd-even alternation can be checked in a magnetic field, since then the odd additional electron may enter in one of two Zeeman states. If the high resistance contact is a tunnel junction, the transmission reflects the density of states. This leads to a direct detection of the so-called Luttinger liquid state of the electrons. Ferromagnetic contacts to the SWCNT leads to a conductance which depends on the orientation of the magnetic domains in the contacts. The magnetoresistance effect can be much larger than expected from a simple spin-valve phenomenon. For any intermediate normal metal (Au) contact resistances, R ∼ h/2e2, the Coulomb blockade may still separate the single electron states in the SWCNT with odd and even numbers of electrons. However, at the lowest temperatures the transmission only shows Coulomb blockade for even number of electrons. In the situations with odd number of electrons a coherent tunnelling process dominates. This shortage of the blockade is rooted in the Kondo states formed in the two Au electrodes by exchange interaction due to the spin state in the SWCNT. This tunnelling process is a result of a net spin on the SWCNT and consequently a spin degeneracy. A triplet state is forced into degeneracy with the singlet state in a suitable magnetic field. The situation in a magnetic field is particularly simple in a SWCNT, in contrast to conventional quantum dots, because the tiny diameter of the SWCNT practically speaking precludes orbital effects.

Strongly interacting Luttinger-liquid state in the integrable model of spinless fermions

A family of spinless fermion models with a hard core potential is formulated and solved by the Bethe ansatz method in one dimension for an arbitrary core radius. The Hamiltonian has two levels of strong interaction: the repulsive hard core potential and the kinetic energy with hopping integrals, values of which depend on the configuration of the particles. The Fermi velocity and the critical exponents describing the asymptotic behavior of the correlation functions at long distances have been calculated numerically for an arbitrary density of electrons and hard core radius. We discuss the effect of the hard core potential. The hard core potential defines an anomalous behavior of the critical exponent \ensuremath{\Theta} of the momentum distribution function. In the high-electron-density region the long-distance behavior is described by a strongly interacting Luttinger-liquid state with \ensuremath{\Theta}>1 and the residual Fermi surface disappears.

Proton spin-lattice relaxation in the critical field region of S=1 Haldane-gap system TMNIN

The temperature dependence of T 1 of the proton in the Haldane-gap system TMNIN has been measured with external fields H up to 7.5 T. For H<H c =2.7 T , the relaxation rate 1/ T 1 decreases with decreasing temperature. These results have been understood in terms of magnon scattering within each of the S z =±1 excited states. For H> H c, on the other hand, 1/ T 1 exhibits a significant increase below about 3 K, which follows the equation 1/ T 1∼ T η−1 with η=0.5, suggesting the realization of a Luttinger-liquid state.

Breakdown of Luttinger liquid state in a one-dimensional frustrated spinless fermion model

The Haldane hypothesis about the universality of the Luttinger liquid (LL) behavior in conducting one-dimensional fermion systems is checked numerically for the spinless fermion model with next-nearest-neighbor interactions. It is shown that for the large enough interactions the ground state can be gapless due to frustrations but at the same time might not belong to the same universality class as a simple LL. The exponents of the correlation functions for this unusual conducting state are found numerically by a finite-size method.

Bounded Luttinger liquids as a universality class of quantum critical behavior

We show that one-dimensional quantum systems with gapless degrees of freedom and open boundary conditions form a universality class of quantum critical behavior, which we propose to call ``bounded Luttinger liquids.'' They share the following properties with ordinary (periodic) Luttinger liquids: the absence of fermionic quasiparticle excitations, charge-spin separation, and anomalous power-law correlations with exponents whose scaling relations are parametrized by a single coupling constant per degree of freedom, ${K}_{\ensuremath{\nu}}.$ The values of ${K}_{\ensuremath{\nu}}$ are independent of boundary conditions, but the representation of the critical exponents in terms of these ${K}_{\ensuremath{\nu}}'\mathrm{s}$ depends on boundary conditions. We illustrate these scaling relations by exploring general rules for boundary critical exponents derived earlier using the Bethe ansatz solution of the one-dimensional Hubbard model together with boundary conformal field theory, and the theory of Luttinger liquids in finite-size systems. We apply this theory to the photoemission properties of the organic conductors $(\mathrm{TMTSF}{)}_{2}X,$ where TMTSF is tetramethyltetraselenafulvalene, and $X={\mathrm{ClO}}_{9},$ ${\mathrm{PF}}_{6},$ $\mathrm{Re}{\mathrm{O}}_{4},$ and discuss to what extent the assumption of finite strands with open boundaries at the sample surface can reconcile the experimental results with independent information on the Luttinger-liquid state in these materials.

Applying voltage sources to a Luttinger liquid with arbitrary transmission

The Landauer approach to transport in mesoscopic conductors has been generalized to allow for strong electronic correlations in a single-channel quantum wire. We describe in detail how to account for external voltage sources in adiabatic contact with a quantum wire containing a backscatterer of arbitrary strength. Assuming that the quantum wire is in the Luttinger liquid state, voltage sources lead to radiative boundary conditions applied to the displacement field employed in the bosonization scheme. We present the exact solution of the transport problem for arbitrary backscattering strength at the special Coulomb interaction parameter g=1/2.

Exactly solvable model of N coupled Luttinger chains

We calculate the exact Green function of a special model of N coupled Luttinger chains with arbitrary interchain hopping t_{perp}. The model is exactly solvable via bosonization if the interchain interaction does not fall off in the direction perpendicular to the chains. For any finite N we find Luttinger liquid behavior and explicitly calculate the anomalous dimension gamma^(N). However, the Luttinger liquid state does not preclude coherent interchain hopping. We also show that gamma^(N) -> 0 for N -> infinity, so that in the limit of infinitely many chains we obtain a Fermi liquid.

Interactions and disorder in multichannel quantum wires

Recent experiments have revealed that the temperature dependence of the conductance of quasi-ballistic quantum wires bears clear features of the Luttinger-liquid state. In this paper, the conductance of an N-channel quantum wire is calculated within the model of N coupled Luttinger liquids and under the assumption of weak disorder. It is shown that as the number of channels increases, a crossover from the Luttinger-liquid to the Fermi-liquid behavior occurs. This crossover manifests itself in the 1/N decrease of the scaling exponent of the temperature dependence. An exact expression for the scaling exponent for the case of N coupled Luttinger chains is obtained, and the large N limit is studied for the case of a quantum wire. The case of N=2 for electrons with spin is analyzed in detail, and a qualitative agreement with the experiment is achieved.

Vicinal Surfaces and the Calogero-Sutherland Model.

A miscut (vicinal) crystal surface can be regarded as an array of meandering but non-crossing steps. Interactions between the steps are shown to induce a faceting transition of the surface between a homogeneous Luttinger liquid state and a low-temperature regime consisting of local step clusters in coexistence with ideal facets. This morphological transition is governed by a hitherto neglected critical line of the well-known Calogero-Sutherland model. Its exact solution yields expressions for measurable quantities that compare favorably with recent experiments on Si surfaces.

Impact of criticality and phase separation on the spin dynamics of the one-dimensional t–J model

The recursion method is used to determine the T=0 spin dynamic structure factor Szz(q,ω) in the Luttinger liquid state and in the phase-separated state of the one-dimensional t–J model. As the exchange coupling increases from zero, the dispersions and line shapes of the dominant spin excitations are observed to undergo a major metamorphosis between the free-fermion limit and the onset of phase separation. The familiar two-spinon spectrum of the Heisenberg antiferromagnetic chain emerges gradually in the strongly phase-separated state.

DISAPPEARANCE OF THE INFRARED CATASTROPHE IN D&gt;1 INTERACTING FERMION SYSTEMS

Within a consistent bosonization treatment in which all the physically-relevant excitations are taken into account, we show explicitly and non-perturbatively for the first time that, for regular interactions, the infrared catastrophe is eliminated by the zero-frequency particle-hole-pair excitations so that the Luttinger liquid state does not appear in dimensions higher than one. The infrared catastrophe is scarce even for systems with singular interactions due to these excitations. We find that the system remains in a Fermi liquid state for the case of unscreened Coulomb interaction.

Consistent bosonization treatment of interacting fermion systems in dimensions higher than one: Requirement for a Luttinger-liquid state.

We present a consistent bosonization treatment for interacting fermion systems in dimensions higher than one. In our analysis, all physically relevant excitations corresponding to forward scattering are included and physical results do not depend on the cutoff parameter introduced in the bosonization process. We calculate the fermion momentum distribution function, from which a condition for the appearance of a Luttinger-liquid state is derived. We find that, for nonsingular interactions, the zero-frequency particle-hole pair excitations directly destroy the orthogonality catastrophe so that there is no Luttinger-liquid fixed point. We further consider the unscreened Coulomb interaction and find that the system remains in a Fermi-liquid state. We observe that the class of singular interactions which are expected to generate a Luttinger-liquid state is very tightly constrained by these zero-frequency excitations.

Universality classes, statistical exclusion principle, and properties of interacting fermions.

We point to the possibility of existence of the statistical-spin-liquid state as the state which differs from either Fermi- or Luttinger-liquid states. In the statistical spin liquid the double occupancies are excluded from the physical space. Each of the above three cases (Fermi, Luttinger, and spin liquids) represents a universality class for the interacting many-particle fermion systems. The properties of the spin liquid such as the chemical potential, the entropy, and the magnetization curve, as well as the quasiparticle structure are briefly discussed.

Randomness at the edge: Theory of quantum Hall transport at filling nu =2/3.

Current Luttinger liquid edge state theories for filling $\nu=2/3$ predict a non-universal Hall conductance, in disagreement with experiment. Upon inclusion of random edge tunnelling we find a phase transition into a new disordered-dominated edge phase. An exact solution of the random model in this phase gives a quantized Hall conductance of 2/3 and a neutral mode propagating upstream. The presence of the neutral mode changes the predicted temperature dependence for tunnelling through a point contact from $T^{2/\nu -2}$ to $T^2$.

Interchain-coupling effect on the one-dimensional spin-1/2 antiferromagnetic Heisenberg model.

Within a mean-field theory, the disordered spin-1/2 single-chain state is shown to be unstable towards an antiferromagnetically, long-range ordered one, when any finite transverse coupling, ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, is introduced between chains. A mean-field Hamiltonian is derived from the Heisenberg model, using an extension of the Wigner-Jordan transformation in two dimensions. The staggered magnetization and the ground-state energy are calculated for values of transverse couplings ranging from 0 to 1. The main consequence of these calculations is that the value of ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$, required to have antiferromagnetic long-range order, must only be nonzero. This result is consistent with that of Sakai and Takahashi indicating that the critical transverse coupling should be of order of the inverse of the one-dimensional staggered magnetic susceptibility. Following a qualitative argument, it can also suggest that the Luttinger-liquid state is unstable at least at nearly half-filling for the Hubbard model in the limit of large Coulomb repulsion.

LUTTINGER-LIQUID BEHAVIOUR IN 2D: THE VARIATIONAL APPROACH

We study a variational formulation of the Luttinger-liquid concept in two dimensions. We show that a Luttinger-liquid wavefunction with an algebraic singularity at the Fermiedge is given by a Jastrow-Gutzwiller type wavefunction, which we evaluate by variational Monte Carlo for lattices with up to 38 × 38 = 1444 sites. We therefore find that, from a variational point of view, the concept of a Luttinger liquid is well defined even in 2D. We also find that the Luttinger liquid state is energetically favoured by the projected kinetic energy in the context of the 2D t-J model. We study and find coexistence of d-wave superconductivity and Luttinger-liquid behaviour in two-dimensional projected wavefunctions. We then argue that generally, any two-dimensional d-wave superconductor should be unstable against Luttinger-liquid type correlations along the (quasi-1D) nodes of the d-wave order parameter, at temperatures small compared to the gap.

Luttinger liquid instability of the 2D t-J model: A variational study.

We study variationally the possible occurrence of a Luttinger liquid in the normal state of the 2D t-J model. For this, we generalize to 2D a Luttinger-Jastrow-Gutzwiller-type wave function introduced by Hellberg and Mele for the 1D t-J model. We show that this wave function does show also in 2D the characteristic correlations of a Luttinger liquid and the gains in kinetic energy stabilize the Luttinger liquid state with respect to Fermi liquid states with short-range correlations only. In addition, we provide rigorous lower bounds to the transition to the fully phase separated state at larger ratios J/t.

Luttinger Liquid in Two Dimensions: Fermionic Backflow and Transfer of Statistics

We discuss certain physical contents of th~ two-dimen~ional Luttinger liquid state using the spin and charge current backflow in strongly correlated electron systems. Also we attempt a description of the Luttinger liquid state in the slave boson formalism. Conven­ tional slave boson mean field theory does not possess an important property expected of a Luttinger liquid, namely a pseudo fermi surface for holons. This arises from the bose condensation of holes. We attempt to rectify this by discussing the physics of the fermionic backflow associated with the hole motion. We argue that the fermionic backflow suppresses the Bose condensation. To understand this we introduce the notion of 'Statistics Transfer' -the medium transfers certain statistics to the slave particle (e's). A variational wave function in which boson condensation is manifestly absent is introduced. The form of the wave function is justified by the concept of statistics transfer. The spinon and holon excita­ tions both exhibiting pseudo fermi surfaces are constructed.