Effective interactions and superconductivity in the t-J model in the large-N limit

The feasibility of a perturbation expansion for Green's functions of the t-J model directly in terms of X-operators is demonstrated using the Baym-Kadanoff functional method. As an application we derive explicit expressions for the kernel of the linearized equation for the superconducting order parameter in leading order of a 1/N expansion. The linearized equation is solved numerically on a square lattice taking instantaneous and retarded contributions into account. Classifying the order parameter according to irreducible representations of the point group C_4v of the square lattice and according to even or odd parity in frequency we find that a reasonably strong instability occurs only for even frequency pairing with d-wavelike symmetry. The corresponding transition temperature T_c is where t is the nearest-neighbor hopping integral. The underlying effective interaction consists of an attractive, instantaneous term and a retarded term due to charge and spin fluctuations. The latter is weakly attractive at low frequencies below ,strongly repulsive up to and attractive towards even higher energies. T_c increases with decreasing doping until a d-wavelike bond-order wave instability is encountered near optimal doping at for J=0.3. T_c is essentially linear in J and rather insensitive to an additional second-nearest neighbor hopping integral t'. A rather striking property of T_c is that it is hardly affected by the soft mode associated with the bond-order wave instability or by the Van Hove singularity in the case with second-nearest neighbor hopping. This unique feature reflects the fact that the solution of the gap equation involves momenta far away from the Fermi surface (due to the instantaneous term) and many frequencies (due to the retarded term) so that singular properties in momentum or frequency are averaged out very effectively. Received: 16 June 1998 / Accepted: 14 July 1998     

Superconductivity in the two-dimensional t-J model

Using computational techniques, it is shown that pairing is a robust property of hole-doped antiferromagnetic insulators. In one dimension and for two-leg ladder systems, a BCS-like variational wave function with long-bond spin singlets and a Jastrow factor provides an accurate representation of the ground state of the t-J model, even though strong quantum fluctuations destroy the off-diagonal superconducting long-range order in this case. However, in two dimensions it is argued-and numerically confirmed using several techniques, especially quantum Monte Carlo-that quantum fluctuations are not strong enough to suppress superconductivity.     

Dimensional crossover in the large-N limit

We consider dimensional crossover for anO(N) Landau-Ginzburg-Wilson model on ad-dimensional film geometry of thicknessL in the large-N limit. We calculate the full universal crossover scaling forms for the free energy and the equation of state. We compare the results obtained using environmentally friendly renormalization with those found using a direct, non-renormalization-group approach. A set of effective critical exponents are calculated and scaling laws for these exponents are shown to hold exactly, thereby yielding nontrivial relations between the various thermodynamic scaling functions.     

Simplified large-N limit in stochastic quantization

In the large-N limit,O(N)-invariant models become exactly soluble due to factorization properties first emphasized by Migdal and Witten. It is shown that in this limit, the Langevin equation of stochastic quantization offers a direct and simple determination of the mass gap. The method is applied to different bosonic and fermionic models.     

On the large-N limit in symplectic matrix models

We study the large-N behavior of SP(N) invariant quantum mechanical matrix models. We establish a saddle-point method through the standard collective field technique and find that it produces the correct large-N behavior. We exhibit, therefore, the semiclassical origin at the large-N limit in this model.     

Approaching the large-N limit: A renormalization group approach to large-N field theories

A technique is developed for finding recursive solutions to field theories that take values in SU(N). The approach can be used as a method of solving the large-N limit as well as calculating finite-N effects. This is illustrated with examples of the anharmonic oscillator, SU(N) chiral model and two-dimensional lattice QCD.     

Wave function(al)s in the large-N limit

The energy spectrum and correlation functions in various quantum theories have been determined within the large-N limit. Here we study the wave functions. Explicit results are presented for a quantum mechanical rotor, invariant under adjoint transformations of U(N), as well as for O(N) invariant vector models in quantum mechanics and quantum field theory.     

Factorisation in large-N limit of lattice gauge theories revisited

We prove using the Schwinger-Dyson equations that the factorisation property holds for all gauge-invariant Green’s function in the large-N limit of a Wilson-Polyakov lattice gauge theory.     

Hadronic decays of η and η′ in the large-N limit

Working in the large-N approximation (N being the number of colors), we relate thestrong η′→ηππ processes to theweak K _L→πππ decays. Chiral corrections are crucial to reproduce the experimental data. The isospin-violating η(η′)→πππ and the weakη→K ⁺π^- decays are also treated in this framework. We comment on the predictions of a “QCD-inspired” approach which determines in principle the chiral symmetry-breaking scales considered, and consequently the Skyrme coupling.