In this paper we describe the electrons of the non-perturbative one-dimensional (1D) Hubbard model by a fluid of unpaired rotated electrons and a fluid of zero-spin rotated-electron pairs. The rotated electrons are related to the original electrons by a mere unitary transformation. For all finite values of energy and for the whole parameter space of the model this two-fluid picture leads to a description of the energy eigenstates in terms of occupancy configurations of η-spin 1/2 holons, spin 1/2 spinons, and c pseudoparticles only. The electronic degrees of freedom couple to external charge (and spin) probes through the holons and c pseudoparticles (and spinons). Our results refer to very large values of the number of lattice sites Na. The holon (and spinon) charge (and spin) transport is made by 2ν-holon (and 2ν-spinon) composite pseudoparticles such that ν = 1, 2, .... For electronic numbers obeying the inequalities N ≤ Na and N ↓ ≤ N ↑ there are no zero-spin rotated-electron pairs in the ground state and the unpaired-rotated-electron fluid is described by a charge c pseudoparticle fluid and a spin ν = 1 two-spinon pseudoparticle fluid. The spin two-spinon pseudoparticle fluid is the 1D realization of the two-dimensional resonating valence bond spin fluid.

We have studied the numerical solutions of Richardson equations of the BCS model in the limit of large number of energy levels at half-filling, and compare them with the analytic results derived by Gaudin and Richardson, which in turn leads to the standard BCS solution. We focus on the location and density of the roots, the eigenvalues of the conserved quantities, and the scaling properties of the total energy for the equally spaced and the two-level models.One of the most studied models in Condensed Matter Physics is the BCS model of superconductivity, proposed by Bardeen, Cooper, and Schrieffer in 1957 [1]. The BCS model is usually formulated in the grand canonical ensemble, which is appropiated to systems with a macroscopic number of fermions. However for small systems, such as nuclei or ultrasmall metallic grains, one has to consider the canonical ensemble where the BCS wave function is not adequate.Fortunately enough, there is a simplified version of the BCS model which is exactly solvable in the canonical ensemble. This occurs when the strengh of the pairing interaction between all the energy levels is the same. The exact solution of the reduced BCS model was obtained by Richardson in 1963 [2, 3, 4], who also studied its consequences and properties in a series of papers in the 60's and 70's [5,6,7,8]. The exactly solvable BCS model, in turn, is closely related to the rational spin model of Gaudin, defined in terms of a set of commuting Hamiltonians [9,10].The integrability of the reduced BCS Hamiltonian was proved in 1997 by Cambiaggio, Rivas, and Saraceno (CRS) [11], who constructed a set of conserved quantities in involution, commuting with the BCS Hamiltonian and closely related to the rational Gaudin's Hamiltonians [9]. More recently Gaudin's trigonometric model have been generalized, to include a g-termà la BCS, in references [12,13]. There are also bosonic pairing Hamiltonians satisfying the same type of equations as in the fermionic case [7,14,15]. Generalizations of the SU (2) BCS model to other Lie groups [16] and supergroups [17] have been worked out. Both, the BCS and Gaudin's models are intimately linked to conformal field theory [18,19,20], Chern-Simons theory [16], and Integrable Models in Statistical Mechanics [21,22,23] (see ref.[24] for a short review on these topics). The exact solution has also been used to study the effect of level statistics [25] and finite temperature [26,27] on the ultrasmall metallic grains.An important property of the exact solution is that the energy of the states and the occupation numbers agree, to leading order in the number of particles, with the BCS theory [1]. This result was obtained by Gaudin [10] and Richardson [8], using an electrostatic analogy of the Richardson's equations, which have to be solved for finding the eigenstates of the BCS Hamiltonian. More recent applications of this electrostatic model to nuclear pairing can be found in ref. [28].In the limit of large number of levels N , Richardson's equations become integral equations, which can be solv...

We show that an extension of the standard BCS Hamiltonian leads to an infinite number of condensates with different energy gaps and self-similar properties, described by a cyclic RG flow of the BCS coupling constant which returns to its original value after a finite RG time.PACS numbers: 11.10. Hi, 74.20.Fg, 71.10.Li The Renormalization Group (RG) continues to be one of the most important tools for studying the qualitative and quantitative properties of quantum field theories and many-body problems in Condensed Matter physics. The emphasis so far has been mainly on flows toward fixed points in the UV or IR. Recently, an entirely novel kind of RG flow has been discovered in a number of systems wherein the RG exhibits a cyclic behavior: after a finite RG transformation the couplings return to their original values and the cycle repeats itself. Thus if one decreases the size of the system by a specific factor that depends on the coupling constants, one recovers the initial system, much like a Russian doll, or quantum version of the Mandelbrot set. Bedaque, Hammer and Van Kolck observed this behavior in a 3-body hamiltonian of interest in nuclear physics [1]. This motivated Glazek and Wilson to define a very simple quantum-mechanical hamiltonian with similar properties [2]. In the meantime such behavior was proposed for a certain regime of anisotropic current-current interactions in 2 dimensional quantum field theory [3].The models in [1,2] are problems in zero-dimensional quantum mechanics, and are thus considerably simpler than the quantum field theory in [3]. In the latter, standard quantum field theory methods of the renormalization group were used, however knowledge of the beta function to all orders was necessary to observe the cyclic flow. What is somewhat surprising is that the model considered in [3] is not very exotic, and is in fact a wellknown theory that arises in many physical problems: at one-loop it is nothing more than the famous KosterlitzThouless RG flow, where the cyclic regime corresponds to |g ⊥ | > |g |. This motivated us to find a simpler many-body problem that captures the essential features of the cyclic RG behavior. We found that a simple extension of the BCS hamiltonian has the desirable properties. Namely, our model is based on the BCS hamiltonian with scattering potential V jj ′ equal to g + iθ for ε j > ε j ′ and g − iθ for ε j < ε j ′ in units of the energy spacing δ.The main features of the spectrum are the following. For large system size, there are an infinite number of BCS condensates, each characterized by an energy gap ∆ n which depends on g, θ. The role of these many condensates becomes clearer when we investigate the RG properties. As in the models considered in [1,2,3], the RG flow possesses jumps from g = +∞ to g = −∞ and a new cycle begins. Let L = e −s L 0 denote the RG scale, which in our problem corresponds to N the number of unperturbed energy levels, and λ the period of an RG cycle:. We show that λ = π/θ. The model we shall consider is an extension of the reduced BC...

We present a systematic construction of effective lagrangians for the low energy and momentum region of ferromagnetic and antiferromagnetic spin waves in crystalline solids. We fully exploit the spontaneous symmetry breaking pattern SU (2) → U (1), the fact that spin waves are its associated Goldstone modes, the crystallographic space group and time reversal symmetries. We show how to include explicit SU (2) breaking terms due to spin-orbit and magnetic dipole interactions. The coupling to electromagnetic fields is also discussed in detail. For definiteness we work with the space group R3c and present our results to next to leading order.