We carry out a complete analytical stability study for the SO(8) symmetric phases in the weakly-interacting two-leg ladder. It is shown that the SO(8) symmetry is robust under generic perturbations. Since there are no fixed points in the one-loop renormalization group equations, the conventional classification of relevant and irrelevant perturbations fails in this case. A new classification is defined and explained in detail. It leads to the anomalous scaling in the ratios of excitation gaps and an universal exponent 1/3 is extracted. This new method is also applied to the well studied 2D Ising model and similar exponent is calculated.
We carry out a complete analytical stability study for the SO(8) symmetric phases in the weakly-interacting two-leg ladder. It is shown that the SO(8) symmetry is robust under generic perturbations. Since there are no fixed points in the one-loop renormalization group equations, the conventional classification of relevant and irrelevant perturbations fails in this case. A new classification is defined and explained in detail. It leads to the anomalous scaling in the ratios of excitation gaps and an universal exponent 1/3 is extracted. This new method is also applied to the well studied 2D Ising model and similar exponent is calculated.
We revisit the problem of interacting electrons hopping on a two-leg ladder. A perturbative renormalization-group analysis reveals that at half-filling the model scales onto an exactly soluble Gross-Neveu model for arbitrary finite-ranged interactions, provided they are sufficiently weak. The Gross-Neveu model has an enormous global SO(8) symmetry, manifest in terms of eight real Fermion fields that, however, are highly nonlocal in terms of the electron operators. For generic repulsive interactions, the two-leg ladder exhibits a Mott insulating phase at half-filling with d-wave pairing correlations. Integrability of the Gross-Neveu model is employed to extract the exact energies, degeneracies, and quantum numbers of all the low-energy excited states, which fall into degenerate SO(8) multiplets. One SO(8) vector includes two charged Cooper pair excitations, a neutral s=1 triplet of magnons, and three other neutral s=0 particle-hole excitations. A triality symmetry relates these eight two-particle excitations to two other degenerate octets, which are comprised of single-electron-like excitations. In addition to these 24 degenerate ``particle'' states costing an energy (mass) m to create, there is a 28-dimensional antisymmetric tensor multiplet of ``bound'' states with energy 3m. Doping away from half-filling liberates the Cooper pairs, leading to quasi-long-range d-wave pair field correlations, but maintaining a gap to spin and single-electron excitations. For very low doping levels, integrability allows one to extract exact values for these energy gaps. Enlarging the space of interactions to include attractive interactions reveals that there are four robust phases possible for the weak coupling two-leg ladder. While each of the four phases has a (different) SO(8) symmetry, they are shown to all share a common SO(5) symmetry---the one recently proposed by Zhang as a unifying feature of magnetism and superconductivity in the cuprates.
Symmetry restoration is usually understood as a renormalization-group induced phenomenon. In this context, the issue of whether one-loop renormalization-group equations can be trusted in predicting symmetry restoration has recently been the subject of much debate. Here we advocate a more pragmatic point of view and expand the definition of symmetry restoration to encompass all situations where the physical properties have only a weak dependence upon an anisotropy in the bare couplings. Moreover we concentrate on universal properties, and so take a scaling limit where the physics is well described by a field theory. In this context, we find a large variety of models that exhibit, for all practical purposes, symmetry restoration: even if symmetry is not restored in a strict sense, physical properties are surprisingly insensitive to the remaining anisotropy. Although we have adopted an expanded notion of symmetry restoration, we nonetheless emphasize that the scaling limit also has implications for symmetry restoration as a renormalization-group induced phenomenon. In all the models we considered, the scaling limit turns out only to permit bare couplings which are nearly isotropic and small. Then the one-loop $\ensuremath{\beta}$ function should contain all the physics and higher loop orders can be neglected. We suggest that this feature generalizes to more complex models. We exhibit a large class of theories with current-current perturbations [of which the SO(8) model of interest in two-leg Hubbard ladders and armchair carbon nanotubes is one] where the one-loop $\ensuremath{\beta}$ functions indicate symmetry restoration; thus we argue that these results can be trusted within the scaling limit.
We analyze an anisotropic spin-1/2 two-leg ladder in the presence of various types of random perturbations. The generic phase diagram for the pure system, in a way similar to spin one chains, consists of four phases: an antiferromagnet, a Haldane gap or a singlet phase (depending on the sign of interchain coupling), and two XY phases designated by XY1 and XY2. The effects of disorder depend crucially on whether it preserves XY symmetry (random field along z and random exchange) or not (random anisotropy or random XY fields). In all cases we computed the new phase diagram and the correlation length for the disordered system. The ladder exhibits a remarkable stability to disorder with XY symmetry. Not only the singlet phases but also the massless XY1 phase are totally unaffected by disorder in stark contrast with the single chain case. Even in the presence of an external magnetic field breaking the spin gap most of the phase diagram (including the XY point) remains unaffected by the disorder, again in opposition with the single chain case. This stabilization towards disorder is similar, albeit stronger, to the one occurring for fermionic ladders. On the other hand, the XY2 phase is very strongly suppressed. Disorder breaking XY symmetry has much stronger effects and most of the phases at the exception of the singlet one are now destroyed. Interestingly, the disordered XY1 phase has a much shorter correlation length than the disordered XY2 phase contrarily with the case of perturbations preserving the rotation symmetry around the z axis. The ladder system is thus in fact much more anisotropic than its single chain counterpart with identical exchanges. Finally we examine the case of strong disorder or weak interchain exchange for XY symmetric disorder. Close to the isotropic point, when the interchain exchange is increased the system undergoes a transition between two decoupled disordered spin-1/2 chains to a singlet ladder unaffected by disorder. For more XY like systems, the transition occurs in two steps with a nondisordered XY1 phase occurring between the decoupled chains and the singlet phase. Comparison with other disordered gapped systems such as spin one chains is discussed.
The phase diagram for the interacting fermions in weak coupling is described by the perturbative renormalization group equations. Due to the lack of analytic solutions for these coupled nonlinear differential equations, it is rather subtle to tell which couplings are relevant or irrelevant. We propose a powerful classification scheme to build up the hierarchy of the relevant couplings by a scaling Ansatz found numerically. To demonstrate its superiority over the conventional classification for the relevant couplings, we apply this scheme to a controversial phase transition in the two-leg ladder and show that it should be a nontrivial crossover instead. The scaling Ansatz we propose here can classify the relevant couplings in hierarchical order without any ambiguity and can improve significantly how we interpret the numerical outcomes in general renormalization group methods.
In previous studies, we proposed a scaling ansatz for electron-electron interactions under renormalization group transformation. With the inclusion of phonon-mediated interactions, we show that the scaling ansatz, characterized by the divergent logarithmic length ld and a set of renormalization-group exponents, also works rather well. The superconducting phases in a doped two-leg ladder are studied and classified by these renormalization-group exponents as demonstration. Finally, nontrivial constraints among the exponents are derived and explained.
We derive two-loop renormalization group equations for the half filled one-dimensional Hubbard chains coupled by the interchain hopping. Our renormalization group scheme for the quasi-one-dimensional electron system is a natural extension of that for the purely one-dimensional systems in the sense that transverse-momentum dependences are introduced in the g-ological coupling constants and we regard the transverse momentum as a patch index. We develop symmetry arguments for the particle-hole symmetric half filled Hubbard model and obtain constraints on the g-ological coupling constants by which resultant renormalization equations are given in a compact form. By solving the renormalization group equations numerically, we estimate the magnitude of excitation gaps and clarify that the charge gap is suppressed due to the interchain hopping but is always finite even for the relevant interchain hopping. To show the validity of the present analysis, we also apply this to the two-leg ladder system. By utilizing the field-theoretical Bosonization and Fermionization method, we derive low-energy effective theory and analyze the magnitude of all the excitation gaps in detail. It is shown that the low-energy excitations in the two-leg Hubbard ladder have $\mathrm{SO}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ symmetry when the interchain hopping exceeds the magnitude of the charge gap.