PHYSICAL REVIEW 8 VOLUME 47, NUMBER 24 15 JUNE 1993-II Conservation laws and correlation functions in the Luttinger liquid Walter Metzner and Carlo Di Castro Dipartimento di Fisica, Universi ta "La Sapienza, "Piazzale A/do Moro 2, 00185 Roma, Italy (Received 30 November 1992) The low-energy properties of interacting Fermi systems are highly constrained by conservation laws. They generally simplify the structure of the underlying renormalization group by reducing the number of independent renormalization constants. In one dimension, all properties of normal metallic fixed points are uniquely determined by separate charge and spin conservation for states near the left and right Fermi points, respectively. We construct the general Luttinger-liquid theory of one-dimensional (1D) metals directly from these conservation laws. Luttinger-liquid parameters emerge naturally from the velocities associated with the conserved currents at the Luttinger-liquid fixed point. Instead of bo- sonization, one may thus use techniques familiar from Fermi-liquid theory, i.e. , Feynman diagrams, equations of motion, and Ward identities. The choice of a technique comprising both Fermi- and Luttinger-liquid theory makes the similarities and dift'erences of both theories particularly transparent, and sets the stage for constructing non-Fermi-liquid metallic fixed points in d & 1. Several generic prop- erties and asymptotic conservation laws of 2D non-Fermi-liquid metals are discussed. I. INTRODUCTION The low-energy behavior of most interacting electron systems has been successfully described by a remarkably small number of "universality" classes. In pure samples and without symmetry breaking, there are only two well- established types of metallic phases, namely the Fermi- liquid phase in d &1 and the Luttinger-liquid phase in d=1. ' There are more if long-range order is allowed, e. g. , superconducting and magnetic phases. Heuristical- ly, this high degree of universality can be understood in terms of Wilson's renormalization group: Close to the Fermi surface, few types of scattering processes sur- vive. ' ' In addition, the structure of the infrared asymp- totic theory is highly constrained by conservation laws. Thus a few basic assumptions determine the entire low- energy behavior in terms of a small set of parameters. Experimental evidence for the breakdown of Fermi- liquid theory in the normal metallic phase of high-T, su- perconductors has led to a search for new, non-Fermi- liquid, metallic states of interacting electrons in d & 1. ' Anderson has pointed out the existence of a singular contribution to the forward-scattering amplitude in two- dimensional Fermi systems, which is missed in standard many-body perturbation theory, but can be obtained by a careful evaluation of the two-particle Schrodinger equa- tion in the presence of a Fermi sea. He argued that this singularity leads to Luttinger-liquid behavior even in d =2. A well-established theory substantiating or falsify- ing this hypothesis is still lacking. The breakdown of Fermi-liquid theory in one- dimensional interacting Fermi systems is already evident in second-order perturbation theory: the perturbative contributions to the quasiparticle weight diverge loga- rithmically at the Fermi surface of the noninteracting system. The problem of treating these divergencies has first been solved by a perturbative renormalization-group approach, known as "g-ology. " Assuming a scaling an- satz for the vertex functions, one approaches the Fermi surface by rescaling the fields and the coupling constants (a small number of "g's"). Depending on the values of the bare couplings, the renormalized couplings Aow ei- ther to strong coupling, and hence out of the perturba- tively controlled regime, or to the exactly soluble Lut- tinger model. In the latter case, the system is a "Lut- tinger liquid, " i.e. , a normal (not symmetry-broken) me- tallic phase characterized by (i) a continuous momentum distribution with a power-law singularity at the Fermi surface, the exponent i) being nonuniversal; (ii) a single- particle density of states which vanishes as cu" near the Fermi energy, i.e. , absence of fermionic quasiparticles; (iii) finite charge- and spin-density response, and the ex- istence of collective bosonic charge- and spin-density modes; (iv) power-law singularities in various supercon- ducting and density correlation functions; and (v) separa- tion of spin and charge degrees of freedom. ' Luttinger-liquid behavior is not confined to special weak-coupling models, but may also govern strongly in- teracting systems. Although in the latter case a perturba- tive calculation of the correlation functions is not ade- quate, the low-energy properties are still uniquely charac- terized by a small number of parameters, which are directly related to simple physical quantities. This leap beyond weak coupling, which follows the spirit of Fermi-liquid theory, has been pioneered by Haldane, who also introduced the suggestive term "Luttinger liquid. " Recently it has been found that the one- dimensional (lD) Hubbard model, which is exactly solu- ble by the Bethe ansatz method, ' is probably a Luttinger liquid for any coupling strength except for half- filling. " ' This conjecture is well established for weak 0163-1829/93/47(24)/16107(17)/$06. 00 16 107 1993 The American Physical Society