We study umklapp couplings and their renormalization-group flow for electrons in a two-dimensional lattice. It is shown that the effective low-energy Hamiltonian involves not only forward scattering, but also considerable nonforward umklapp scattering, when the electrons fill a substantial fraction of the valence band. The behavior of these couplings under the renormalization group is studied in the loop expansion. It is shown that they remain marginal. We conclude with the possible consequences of this on properties of such electronic systems and their Fermi-liquid theory. \textcopyright{} 1996 The American Physical Society.
The role of Umklapp scattering for electrons on a lattice is studied. Using renormalisation group methods it is shown that the effective low energy interaction involves not only forward scattering, but also non-forward Umklapp scattering. We show that the latter can happen to almost all of the electrons even when they have a filling of around one-half. The bahavior of these Umklapp couplings is studied in the loop expansion. It is shown that they remain marginal to all loop orders. The discussion is given in the context of two-dimensional lattices, but generalisation to three-dimensions is mentioned. We also briefly discuss the possible consequences of this on Fermi liquid theory and on physical properties of such electronic systems.
An exact solution of the one-dimensional electron gas on a lattice is given for the special case in which two of the four coupling constants have particular values. It is shown that umklapp scattering has the same effect on the charge-density waves as backward scattering across the Fermi "surface" has on the spin-density waves and that the method of Luther and Emery can be used to solve this more general problem. For repulsive electron-electron interactions the umklapp scattering produces a gap in the charge-density wave spectrum and this appears for low-lying excitations when there is a half-filled band. The use of renormalization-group scaling to solve for general values of the coupling constants is discussed.
We study the renormalization-group (RG) flow of interactions in the two-dimensional ${t\ensuremath{-}t}^{\ensuremath{'}}$ Hubbard model near half-filling in an N-patch representation of the whole Fermi surface. Starting from weak to intermediate couplings the flows are to strong coupling, with different characters depending on the choice of parameters. In a large parameter region elastic umklapp scatterings drive an instability which on parts of the Fermi surface exhibits the key signatures of an insulating spin liquid (ISL), as proposed by Furukawa, Rice, and Salmhofer [Phys. Rev. Lett. 81, 3195 (1998)] rather than a conventional symmetry-broken state. The ISL is characterized by both strong d-wave pairing and antiferromagnetic correlations; however, it is insulating due to the vanishing local charge compressibility and a spin liquid because of the spin gap arising from the pairing correlations. We find that the unusual RG flow, which we interpret in terms of an ISL, is a consequence of a Fermi surface close to the saddle points at the Brillouin-zone boundaries which provides an intrinsic and mutually reinforcing coupling between pairing and umklapp channels.
We review the theory of interacting Fermi systems whose low-energy physics is dominated by forward scattering, that is 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 behaviour 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 lowenergy theories. Particular attention is paid 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.
We show that to understand the orthogonality catastrophe in the half-filled lattice model of spinless fermions with repulsive nearest-neighbor interaction and a local impurity in its Luttinger liquid phase one has to take into account (i) the impurity scaling, (ii) unusual finite-size
L corrections of the form
ln(L)/L, as well as (iii) the renormalization group flow of the umklapp scattering. The latter defines a length scale
Lu, which becomes exceedingly large the closer the system is to its transition into the charge density wave phase. Beyond this transition umklapp scattering is relevant in the renormalization group sense. Field theory can only be employed for length scales larger than
Lu. For small to intermediate two-particle interactions, for which the regime
L>Lu can be accessed, and taking into account the finite-size corrections resulting from (i) and (ii) we provide strong evidence that the impurity backscattering contribution to the orthogonality exponent is asymptotically given by
1/16. While further increasing the two-particle interaction leads to a faster renormalization group flow of the impurity towards the cut chain fixed point, the increased bare amplitude of the umklapp scattering renders it virtually impossible to confirm the expected asymptotic value of
1/16 given the accessible system sizes. We employ the density matrix renormalization group.