Using a Luttinger liquid theory we investigate the time evolution of the particle density of a one-dimensional spinful fermionic system with open boundaries and subject to a finite-duration quench of the inter-particle interaction. Taking into account also the turning on of an umklapp perturbation to the system Hamiltonian as a result of the quench, we study the possible formation of a Wigner molecule inside the system, focusing in particular on the sudden and adiabatic regimes. We show that the creation of this correlated state is essentially due to the propagation of "light-cone" perturbations through system which arise after both switching on and switching off the quenching protocol and that its behavior strongly depends on the quench duration.
We study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liquid. For a range of repulsive interactions, we find that for symmetric barriers there exist resonances with a universal peak conductance $2g^* e^2/h$ that are controlled by a non-trivial intermediate fixed point. This fixed point is also a quantum critical point separating symmetry-protected topological phases. By tuning the system through resonance, all SPT phases can be accessed. For a particular interaction strength with Luttinger parameters $g_ρ=1/3$ and $g_σ=1$, we show that the problem is equivalen…
We introduce the concept of a bosonic spin liquid condensate (SLC), where spinful bosons in a lattice form a zero-temperature spin disordered charge condensate that preserves the spin rotation symmetry, but breaks the U(1) symmetry due to a spinless order parameter with charge one. It has an energy gap to all the spin excitations. We show that such SLC states can be realized in a system of spin $S\ge 2$ bosons. In particular, we analyze the SLC phase diagram in the spin 2 case using a mean-field variational wave function method. We show there is a direct analogy between the SLC and the resonat…
We study the influence of spin on the quantum interference of interacting electrons in a single-channel disordered quantum wire within the framework of the Luttinger liquid (LL) model. The nature of the electron interference in a spinful LL is particularly nontrivial because the elementary bosonic excitations that carry charge and spin propagate with different velocities. We extend the functional bosonization approach to treat the fermionic and bosonic degrees of freedom in a disordered spinful LL on an equal footing. We analyze the effect of spin-charge separation at finite temperature both o…
We calculate the finite temperature single hole spectral function and the spin dynamic structure factor of spinfull one-dimensional Tomonaga-Luttinger liquid. Analytical expressions are obtained for a number of special cases. We also calculate the single hole spectral function of a spin gapped Luther-Emery liquid and obtain exact results at the free fermion point K_s=1/2. These results may be applied to the analysis of angle resolved photoemission and neutron scattering experiments on quasi-one-dimensional materials.
The transport dynamics of a quenched Luttinger liquid tunnel-coupled to a fermionic reservoir is investigated. In the transient dynamics, we show that for a sudden quench of the electron interaction universal power-law decay in time of the tunneling current occurs, ascribed to the presence of entangled compound excitations created by the quench. In sharp contrast to the usual non universal power-law behavior of a zero-temperature non-quenched Luttinger liquid, the steady state tunneling current is ohmic and can be explained in terms of an effective quench-activated heating of the system. Our s…
Effects of a single impurity potential on bulk properties of a spinful Tomonaga-Luttinger (TL) liquid is studied. A boundary bosonization technique is developed to include the impurity potential of {\it arbitrary} strength $V$. Our new bosonization formula for fermion field $ψ$ smoothly connects the two existing expressions in the strong ($V = \infty$) and the weak ($V = 0$) impurity limits. With use of the formula, we found the TL parameters determined from the long-distance correlation functions are renormalized due to the partial transmission through the impurity potential.
We study the dynamics resulting out of an abrupt change of the two-particle interaction in two models of closed one-dimensional Fermi systems: (a) the field theoretical Tomonaga-Luttinger model and (b) a microscopic lattice model. Using a nonperturbative approach which is controlled for small two-particle interactions we are able to reach large times allowing us to access the properties of the steady state of the lattice model. Comparing those to the exact solution of the full dynamics in the Tomonaga-Luttinger model we provide evidence for universal Luttinger liquid behavior.
We study the non-equilibrium dynamics of a Luttinger liquid after a simultaneous quantum quench of the interaction and dissipative quench to the environment within the realm of the Lindblad equation. When the couplings to environment satisfy detailed balance, the system is destined to thermalize, which we follow using bosonization. The thermodynamic entropy of the system, which also encodes information about the entanglement with the environment, either exhibits a maximum and minimum or grows monotonically before reaching its thermal value in a universal fashion. The single-particle density ma…
We investigate a one dimensional quantum fluid coupled to a dissipative bath. The quantum fluid is captured by the canonical Luttinger liquid; the bath is given by the model of Caldeira and Leggett, i.e. a tower of oscillators coupled linearly to the fluid density, $ρ$. The bath can be integrated out exactly, producing an effective interaction for the fluid that is nonlocal in time; we argue that the form corresponding to Ohmic dissipation is generic. Compared to previous works, we compute correlation functions for this minimal model without approximation, including at finite temperature $T>0$…
We investigate the decoherence of the electron wavepacket in purely ballistic one-dimensional systems described through the Luttinger liquid (LL). At a finite temperature $T$ and long times $t$, we show that the electron Green's function for a fixed wavevector close to one Fermi point decays as $\exp(-t/τ_F)$, as opposed to the power-law behavior occurring at short times, and the emerging electron lifetime obeys $τ_F^{-1}\propto T$ for spinful as well as spinless electrons. For strong interactions, $(Tτ_F)\ll 1$, reflecting that the electron is not a good Landau quasiparticle in LLs. We justif…
One of the most promising proposals for engineering topological superconductivity and Majorana fermions employs a spin-orbit coupled nanowire subjected to a magnetic field and proximate to an s-wave superconductor. When only part of the wire's length contacts to the superconductor, the remaining conducting portion serves as a natural lead that can be used to probe these Majorana modes via tunneling. The enhanced role of interactions in one dimension dictates that this configuration should be viewed as a superconductor-Luttinger liquid junction. We investigate such junctions between both helica…
We study how a Luttinger liquid of spinless particles in one dimension approaches thermal equilibrium. Full equilibration requires processes of backscattering of excitations which occur at energies of order of the bandwidth. Such processes are not accounted for by the Luttinger liquid theory. We treat the high-energy excitations as mobile impurities and derive an expression for the equilibration rate in terms of their spectrum. Our results apply at any interaction strength.
We provide analytic expressions for the Green's functions in position-frequency space as well as for the tunneling density of states of various Luttinger liquids at zero temperature: the standard spinless and spinful Luttinger liquids, the helical Luttinger liquid at the edge of a topological insulator, and the Luttinger liquid that appears either together with an ordering transition of nuclear spins in a one-dimensional conductor, or in spin-orbit split quantum wires in an external magnetic field. The latter system is often used to mimic a helical Luttinger liquid, yet we show here that it ex…
We describe a new class of spin liquids with global SU(2) spin rotation symmetry in spin 1/2 systems on the triangular lattice, which have real Majorana fermion excitations carrying spin S = 1. The simplest translationally-invariant mean-field state on the triangular lattice breaks time-reversal symmetry and is stable to fluctuations. It generically possesses gapless excitations along 3 Fermi lines in the Brillouin zone. These intersect at a single point where the excitations scale with a dynamic exponent z = 3. An external magnetic field has no orbital coupling to the SU(2) spin rotation-inva…
We derive generalized Kronig identities expressing quadratic fermionic terms including momentum transfer to bosonic operators and use them to obtain the exact solution for one-dimensional fermionic models with linear dispersion in the presence of position-dependent interactions and scattering potential. In these Luttinger droplets, which correspond to Luttinger liquids with spatial variations or constraints, the position dependences of the couplings break the translational invariance of correlation functions and modify the Luttinger-liquid interrelations between excitation velocities.
We have devised a general scheme that provides exact results for the scaling dimensions of all allowed in multi-channel Luttinger Liquids local perturbations. The technique developed provides universal description and establishes relations between scaling dimensions of perturbations in different phases of Luttinger Liquid. These multiple relations between scaling dimensions are universal. They are valid for all Luttinger Liquids with arbitrary current-current and density-density interactions. We have also established duality transformation between different subsets of Luttinger Liquid phases k…
We present the theoretical basis for analyzing resonant Raman scattering experiments in one-dimensional systems described by the Luttinger liquid fixed point. We make experimentally testable predictions for distinguishing Luttinger liquids from Fermi liquid and argue that presently available quantum wire systems are \textit{not} in the regime where Luttinger liquid effects are important.
We give a brief introduction to Luttinger liquids and to the phenomena of electronic transport or conductance in quantum wires. We explain why the subject of transport in Luttinger liquids is relevant and fascinating and review some important results on tunneling through barriers in a one-dimensional quantum wire and the phenomena of persistent currents in mesoscopic rings. We give a brief description of our own work on transport through doubly-crossed Luttinger liquids and transport in the Schulz-Shastry exactly solvable Luttinger-like model.
We study a Luttinger Liquid in a finite one-dimensional wire with box-like boundary conditions by considering the local distribution of the single particle spectral weight. This corresponds to the experimental probability of extracting a single electron at a given place and energy, which can be interpreted as the square of an electron wave-function. For the non-interacting case, this is given by a standing wave at the Fermi wave-vector. In the presence of interactions, however, the wave-functions obtain additional structure with a sharp depletion near the edges and modulations throughout the w…
We calculate the correlation functions and the DC conductivity of Luttinger liquid superlattices, modeled by a repeated pattern of interacting and free Luttinger liquids. In a specific realization, where the interacting subsystem is a Hubbard chain, the system exhibits a rich phase diagram with four different phases: two metals and two compressible insulators. In general, we find that the effective low energy description amalgamates features of both types of liquids in proportion to their spatial extent, suggesting the interesting possibility of `engineered' Luttinger liquids.
We show that the one-dimensional (1D) electron systems can also be described by Landau's phenomenological Fermi-liquid theory. Most of the known results derived from the Luttinger-liquid theory can be retrieved from the 1D Fermi-liquid theory. Exact correspondence between the Landau parameters and Haldane parameters is established. The exponents of the dynamical correlation functions and the impurity problem are also discussed based on the finite size corrections of elementary excitations with the predictions of the conformal field theory, which provides a bridge between the 1D Fermi-liquid an…
This short piece celebrates Haldane's seminal J. Phys. C 14, 2585 (1981) paper laying the foundations of the modern theory of Luttinger liquids in one-dimensional systems.
This chapter reviews the theoretical description of interacting fermions in one dimension. The Luttinger liquid concept is elucidated using the Tomonaga-Luttinger model as well as integrable lattice models. Weakly coupled chains and attempts to experimentally verify the theoretical predictions are discussed.
In these lecture notes, the basic physics of Fermi liquids and Luttinger liquids is presented. Fermi liquids are discussed both from a phenomenological viewpoint, in relation to microscopic approaches, and as renormalization group fixed points. Luttinger liquids are introduced using the bosonization formalism, and their essential differences with Fermi liquids are pointed out. Applications to transport effects, the effect of disorder, quantum spin chains, and spin ladders, both insulating and metallic, are given.
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 $\mathbf{a}_{\mathbf{k}}$ 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 study the problem of Luttinger liquids interacting with an active environment. We are particularly interested in how dissipation affects the response and correlation functions of non-isolated Luttinger liquids. We show that the exchange of particles, energy, and momentum lead to changes in the exponents characterizing the various correlations functions. We discuss the importance of the zero mode physics in this context.
We study a simple model of a quantum Hall system with the electrons confined to a linear, narrow channel. The system is mapped to a 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. We study this mapping at the microscopic level, and discuss the relation between the parameters of the 2D system and the corresponding 1D Luttinger liquid parameters. We focus in particular on how the parameters are renormalized by the electron interactions and show that the velocity parameter of the…
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.
Hidden duality and associated instabilities of the spinless Luttinger liquid on lattice are reported. The local quantum fluctuations 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 $θ$-term and with the rich hidden duality.
We generalize nonlinear Luttinger liquid theory to describe the dynamics of one-dimensional quantum critical systems at low temperatures. Analyzing density-matrix renormalization group results for the spin autocorrelation function in the XXZ chain we provide, in particular, direct evidence for spin diffusion in sharp contrast to the exponential decay in time predicted by conventional Luttinger liquid theory. Furthermore, we discuss how the frequencies and exponents of the oscillatory contributions from the band edges are renormalized by irrelevant interactions and obtain excellent agreement be…
We study the effect of spin injection into a Luttinger liquid. The spin-injection-detection setup of Johnson and Silsbee is considered; here spins injected into the Luttinger liquid induce, across an interface with a ferromagnetic metal, either a spin-dependent current ($I_s$) or a spin-dependent boundary voltage ($V_s$). We find that the spin-charge separation nature of the Luttinger liquid affects $I_s$ and $V_s$ in a very different fashion. In particular, in the Ohmic regime, $V_s$ depends on the spin transport properties of the Luttinger liquid in essentially the same way as it would in th…
It is shown theoretically that the Luttinger liquid phase in quasi-one-dimensional conductors can exist in the presence of impurities in a form of a collection of bounded Luttinger liquids. The conclusion is based upon 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 co…
We investigate tunneling between two spinful Tomonaga-Luttinger liquids (TLLs) realized, e.g., as two crossed nanowires or quantum Hall edge states. When injecting into each TLL one electron of opposite spin, the dc current measured after the crossing differs for singlet, triplet, or product states. This is a striking new non-Fermi liquid feature because the (mean) current in a noninteracting beam splitter is insensitive to spin entanglement. It can be understood in terms of collective excitations subject to spin-charge separation. This behavior may offer an easier alternative to traditional e…
It is well-known that, generically, the one-dimensional interacting fermions cannot be described in terms of the Fermi liquid. Instead, they present different phenomenology, that of the Tomonaga-Luttinger liquid: the Landau quasiparticles are ill-defined, and the fermion occupation number is continuous at the Fermi energy. We demonstrate that suitable fine-tuning of the interaction between fermions can stabilize a peculiar state of one-dimensional matter, which is dissimilar to both the Tomonaga-Luttinger and Fermi liquids. We propose to call this state a quasi-Fermi liquid. Technically speaki…
We consider a system of 2D interacting fermions with a flat Fermi surface. The apparent conflict between Luttinger and non Luttinger liquid behavior found through different approximations is resolved by showing the existence of a line of non trivial fixed points, for the RG flow, corresponding to Luttinger liquid behavior; the presence of marginally relevant operators can cause flow away from the fixed point. The analysis is non-perturbative and based on the implementation, at each RG iteration, of Ward Identities obtained from local phase transformations depending on the Fermi surface side, i…
The influences of an impurity on the spin and the charge transport of one-dimensional antisymmetric spin filter are investigated using bosonization and Keldysh formulation and the results are highlighted against those of spinful Luttinger liquids. Due to the dependence of the electron spin orientation on wave number the spin transport is not affected by the impurity, while the charge transport is essentially identical with that of spinless one-dimensional Luttinger liquid.
I give a brief introduction to Luttinger liquids. Luttinger liquids are paramagnetic one-dimensional metals without Landau quasi-particle excitations. The elementary excitations are collective charge and spin modes, leading to charge-spin separation. Correlation functions exhibit power-law behavior. All physical properties can be calculated, e.g. by bosonization, and depend on three parameters only: the renormalized coupling constant $K_ρ$, and the charge and spin velocities. I also discuss the stability of Luttinger liquids with respect to temperature, interchain coupling, lattice effects and…
We demonstrate that the notion of spin-charge separation as follows from the Tomonaga-Luttinger theory of relativistic fermions does not suffice to completely specify the Luttinger liquid state associated with the fermions of condensed matter physics. The latter carries in addition a form of topological order which can be made visible using non-local correlation functions familiar from lattice gauge theory.
We provide strong evidence that the relaxation dynamics of one-dimensional, metallic Fermi systems resulting out of an abrupt amplitude change of the two-particle interaction has aspects which are universal in the Luttinger liquid sense: The leading long-time behavior of certain observables is described by universal functions of the equilibrium Luttinger liquid parameter and the renormalized velocity. We analytically derive those functions for the Tomonaga-Luttinger model and verify our hypothesis of universality by considering spinless lattice fermions within the framework of the density matr…
In quasi-1D conducting nanowires spin-orbit coupling destructs spin-charge separation, intrinsic to Tomonaga-Luttinger liquid (TLL). We study renormalization of a single scattering impurity in a such liquid. Performing bosonization of low-energy excitations and exploiting perturbative renormalization analysis we extend the phase portrait in $K_s$ - $K_c$ space, obtained previously for TLL with decoupled spin-charge channels.
The spinless fermion model with hard core repulsive potential extended on a few lattice sites is considered.The Luttinger liquid behaviour is studied for the different values of a hard core radius. A critical exponent of the one particle correlation function for arbitrary electron density and coupling constant are obtained.
The concept of a Luttinger liquid has recently been established as a fundamental paradigm vital to our understanding of the properties of one-dimensional quantum systems, leading to a number of theoretical breakthroughs. Now theoretical predictions have been put to test by the comprehensive experimental study.
It is shown how Luttinger liquids may be studied using sea-bosons. The main advantage of the sea-boson method is its ability to provide information about short-wavelength physics in addition to the asymptotics and is naturally generalisable to more than one dimension. In this article, we solve the Luttinger model and the Calogero-Sutherland model, the latter in the weak-coupling limit. The anomalous exponent we obtain in the former case is identical to the one obtained by Mattis and Lieb. We also apply this method to solve the two-dimensional analog of the Luttinger model and show that the sys…
We develop the renormalization group theory of the conductances of N-lead junctions of spinless Luttinger-liquid wires as functions of bias voltages applied to N independent Fermi-liquid reservoirs. Based on the perturbative results up to second order in the interaction we demonstrate that the conductances obey scaling. The corresponding renormalization group $β$ functions are derived up to second order.
We consider a Laughlin droplet in a confining potential which is very steep but also weak compared to the ultra-short ranged inter-particle interactions. We find that the eigenstates have a Jack polynomial structure, and have an energy spectrum which is extremely different from the well-known Luttinger liquid edge.
The exact Green's functions of a finite-size Luttinger Liquid (LL) connected to impedances are computed at zero and finite temperature. Bosonization for a LL with Impedance boundary conditions (IBC) is proven to hold. The LL with open boundary conditions (for both Neumann and Dirichlet cases) is explicitly recovered as a special limit when one has infinite impedances. Additionally when the impedances are equal to the characteristic impedance of the Luttinger liquid then the finite Luttinger liquid is shown to be effectively equivalent to an infinite Luttinger liquid.
We study the Kondo effect in two crossed Luttinger liquids, using Boundary Conformal Field Theory. We predict two types of critical behaviors: either a two-channel Kondo fixed point with a nonuniversal Wilson ratio, or a new theory with an anomalous response identical to that found by Furusaki and Nagaosa (for the Kondo effect in a single Luttinger liquid). Moreover, we discuss the relevance of perturbations like channel anisotropy, and we make links with the Kondo effect in a two-band Hubbard system modeled by a channel-dependent Luttinger Hamiltonian. The suppression of backscattering off th…
We show using bosonization techniques, that strong forward scattering interactions between one dimensional spinless Luttinger liquids (LL) can stabilize a phase where charge-density wave, superconducting and transverse single particle hopping perturbations are irrelevant. This new phase retains its LL like properties in the directions of the chains, but with relations between exponents modified by the transverse interactions, whereas, it is a perfect insulator in the transverse direction. The mechanism that stabilizes this phase are strong transverse charge density wave fluctuations at incomme…
We report the detailed non-Fermi liquid (NFL) lineshape of the dispersing excitation which defines the Fermi surface (FS) for quasi-one-dimensional Li_{0.9}Mo_{6}O_{17}. The properties of Li_{0.9}Mo_{6}O_{17} strongly suggest that the NFL behavior has a purely electronic origin. Relative to the theoretical Luttinger liquid lineshape, we identify significant similarities, but also important differences.
We have developed a fermionic boot-strap method to solve a class of chiral one dimensional fermion model which cannot be tackled by bosonization. Using this scheme, we show that Luttinger liquid behavior in a gas of four interacting chiral Majorana fermions is highly sensitive to the velocity degeneracy. Upon changing the velocity of a one chiral fermion, a sharp bound (or anti-bound) state splits off from the original Luttinger liquid continuum, cutting off the X-ray singularity to form a broad incoherent excitation with a lifetime that grows linearly with frequency.
We report the exact calculation of the Fermi-edge singularity exponent for correlated electrons in one dimension (Luttinger liquid). Focusing on the special interaction parameter g=1/2, the asymptotic long-time behavior can be obtained using the Wiener-Hopf method. The result confirms the previous assumption of an open boundary fixed point. In addition, a dynamic k-channel Kondo impurity is studied via Abelian bosonization for k=2 and k=4. It is shown that the corresponding orthogonality exponents are related to the orthogonality exponent in a Luttinger liquid.
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 s…
One-dimensional electrons with a linearized dispersion relation are equivalent to a collection of harmonic plasmon modes, which represent long wavelength density oscillations. An immediate consequence of this Luttinger model of one-dimensional electron systems is the absence of inelastic scattering processes responsible for the relaxation of nonequilibrium states. In a generic nonlinear Luttinger liquid plasmons may decay and thus acquire a finite lifetime. We show that equilibration of plasmons is hierarchical and has profound implications for the dynamics after a thermal quench. We also deve…
We study systems of coupled spin-gapped and gapless Luttinger liquids. First, we establish the existence of a sliding Luttinger liquid phase for a system of weakly coupled parallel quantum wires, with and without disorder. It is shown that the coupling can {\it stabilize} a Luttinger liquid phase in the presence of disorder. We then extend our analysis to a system of crossed Luttinger liquids and establish the stability of a non-Fermi liquid state: the crossed sliding Luttinger liquid phase (CSLL). In this phase the system exhibits a finite-temperature, long-wavelength, isotropic electric cond…
We show that the Luttinger theorem, a robust feature of Fermi liquids, can be violated in non-Fermi liquids. We compute non-Fermi liquid Green functions using duality to black holes and find that the volume of the Fermi surface depends exponentially on the scaling dimension, which is a measure of the coupling. This demonstrates that Luttinger's theorem does not extend to non-Fermi liquids. We comment on possible experimental signatures.
We show that claims, presented in Phys. Rev. B {\bf 65}, 125109 (2002) concerning the threshold behavior of dynamical correlations of a Coulomb Luttinger liquid, are based on an inconsistent mathematical analysis. Physical arguments are also presented, indicating that the claimed behavior is unlikely to be correct.
In a recent Physical Review Letter, Oreg and Finkel'stein (OF) have calculated the electron density of states (DOS) for tunneling into a repulsive Luttinger liquid close to the location of an impurity. The result of their calculation is a DOS which is enhanced with respect to the pure system, and moreover diverging for not too strong repulsion. In this Comment we intend to show that OF's calculation suffers from a subtle flaw which, being corrected, results into a DOS not only vanishing at zero frequency but in fact suppressed in comparison with the DOS of a pure Luttinger liquid.
Two one-dimensional Luttinger liquid systems coupled by Hund's coupling are studied by the renormalization group and the non-abelian bosonization methods. It is found that the Hund's coupling is always relevant irrespective of the repulsive interaction between electrons within each chain. The properties of the resulting strong coupling fixed point are discussed.
