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arXiv:cond-mat/9801285 · Physical review. B, Condensed matter

Exact SO(8) symmetry in the weakly-interacting two-leg ladder

Hsiu‐Hau Lin, Leon Balents, Matthew P. A. Fisher

Published Jul 15, 1998• 189 citations
arXivHTMLPDF

Abstract

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 $\sqrt{3}m.$ 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.

DOI
10.1103/physrevb.58.1794

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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 ddd-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=1s=1s=1 triplet of magnons, and three other neutral s=0s=0s=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.\sqrt{3}m.3​m. Doping away from half-filling liberates the Cooper pairs, leading to quasi-long-range ddd-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.
Recently Lin, Balents, and Fisher have demonstrated that two-leg Hubbard ladders and armchair carbon nanotubes renormalize towards the integrable SO(8) Gross-Neveu model. Here, we exploit this integrability to extract non-trivial effects of interactions in these systems in their doped phase. We so obtain results that could not be expected a priori. Using thermodynamic Bethe ansatz, we compute exactly both the spin and single-particle gaps and the Luttinger parameter describing low-energy excitations. We show rigourously both the spin gap and the electron gap do not vanish at finite doping, while the Luttinger parameter remains close to its free Fermionic value of 1, even for larger values of doping. A similar set of conclusions is drawn for the undoped systems' behavior in a finite magnetic field. We also comment on the existence in these systems of the $\ensuremath{\pi}$ resonance, a hallmark of Zhang's SO(5) theory of high-Tc{T}_{c}Tc​ superconductivity.
Motivated by recent work of Lin, Balents, and Fisher [Phys. Rev. B 58, 1794 (1998)], we compute correlation functions at zero temperature for weakly coupled two-leg Hubbard ladders and (N,N)(N,N)(N,N) armchair carbon nanotubes. In this paper it was argued that such systems renormalize towards the SO(8) Gross-Neveu model, an integrable theory. We exploit this integrability to perform the computation at the SO(8) invariant point. Any terms breaking the SO(8) symmetry can be treated systematically in perturbation theory, leading to a model with the same qualitative features as the integrable theory. Using said correlators, we determine the optical conductivity, the single-particle spectral function, and the $I\ensuremath{-}V$ curve for tunneling into the system from an external metallic lead. The frequency, \ensuremath{\omega}, dependent optical conductivity is determined exactly for $\ensuremath{\omega}<3m$ (m(m(m being the fermion particle mass in the SO(8) Gross-Neveu model). It is characterized by a sharp ``exciton'' peak at $\ensuremath{\omega}=\sqrt{3}m,$ followed by the onset of the particle-hole continuum beginning at $\ensuremath{\omega}=2m.$ Interactions modify this onset to $\ensuremath{\sigma}(\ensuremath{\omega}+2m)\ensuremath{\sim}{\ensuremath{\omega}}^{1/2}$ and not the ${\ensuremath{\omega}}^{\ensuremath{-}1/2}$ one would expect from the van Hove singularity in the density of states. Similarly, we obtain the exact single-particle spectral function for energies less than 3m.3m.3m. The latter possesses a $\ensuremath{\delta}$ function peak arising from single-particle excitations, together with a two-particle continuum for $\ensuremath{\omega}>~2m.$ The final quantity we compute is the tunneling $I\ensuremath{-}V$ curve to lowest nonvanishing order in the tunneling matrix elements. For this quantity, we present exact results for voltages, V<(1+3)m.V<(1+\sqrt{3})m.V<(1+3​)m. The resulting differential conductance is marked by a finite jump at $\ensuremath{\omega}=2m,$ the energy of the onset of tunneling into the continuum of two-particle states. Through integrability, we are able to characterize this jump exactly. All calculations are done through form-factor expansions of correlation functions. These give exact closed form expressions for spectral functions because the SO(8) Gross-Neveu model is massive: each term in the expansion has an energy threshold below which it does not contribute. Thus, we obtain exact results below certain thresholds by computing a finite number of terms in this series. Previous to this paper, the only computed form factor of SO(8) Gross-Neveu was the two-particle form factor of an SO(8) current with two fundamental fermions. In this paper we compute the set of all one- and two-particle form factors for all relevant fields, the currents as well as the kinks and fermions.
We study the dimensional crossover from one to two dimensions in half-filled and lightly doped, weakly interacting N-leg Hubbard ladders. In this case, the Hubbard ladders are equivalent to an N-band model. Using renormalization-group techniques, we find that the half-filled ladders exhibit (in the spin sector) an odd-even effect only below a crossover energy ${E}_{c}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\ensuremath{\alpha}\mathrm{exp}(\ensuremath{\gamma}N)]$ $(\ensuremath{\alpha}\ensuremath{\ll}1$ and $\ensuremath{\gamma}\ensuremath{\sim}1$ depend on the interaction strength and on the hopping matrix elements): Below Ec,{E}_{c},Ec​, the dominant interactions take place within band pairs $(j,N+1\ensuremath{-}j)$ [and within the band (N+1)/2(N+1)/2(N+1)/2 for N odd], such that even-leg ladders are an insulating spin liquid, while odd-leg ladders have one gapless spinon mode. In contrast, above the energy scale Ec,{E}_{c},Ec​, all bands are interacting with each other and the system is a two-dimensional- (2D)-like (insulating) antiferromagnet; we obtain an analytical expression for the Hamiltonian which is similar to the 2D Heisenberg antiferromagnet. Bosonization techniques show that in the charge sector the Mott insulator is as well below and above Ec{E}_{c}Ec​ of the same type as in N/2N/2N/2 half-filled two-leg ladders. Doping away from half-filling, we find that the effect of an increasing doping $\ensuremath{\delta}$ is very similar to decreasing the number of legs N: In both cases interactions between unpaired bands are suppressed and thereby the antiferromagnetic correlations reduced. The resulting band pairs form then insulating spin liquids and when doped, there is a spin gap, but phase coherence exists only within the band pairs. At higher doping levels ${\ensuremath{\delta}}_{c}={\ensuremath{\delta}}_{c}(N),$ phase coherence between all band pairs sets in and the system becomes a 2D-like d-wave superconductor $[{\ensuremath{\delta}}_{c}(N)\ensuremath{\rightarrow}0$ for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}].$
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.
Inspired by the observation of a robust ddd-wave superconducting phase driven by tuning the next-nearest-neighbor (NNN) electron hopping in recent density matrix renormalization group (DMRG) studies of six- and eight-leg $t\text{\ensuremath{-}}J$ model, we systematically study the phase diagram of the two-leg $t\text{\ensuremath{-}}J$ ladder with the NNN couplings (including NNN hopping and spin interaction) in a large region of doping level, by means of the DMRG calculations. Upon doping from half filling, we identify the Luther-Emery liquid (LEL) phase, which can be distinguished as the pairing-dominant and charge density-dominant regime by comparing the Luttinger parameter ${K}_{\ensuremath{\rho}}$. With the growing NNN couplings, pairing correlations are enhanced and correspondingly ${K}_{\ensuremath{\rho}}$ increases, driving the system from the density-dominant to the pairing-dominant regime. In the Tomonaga-Luttinger liquid (TLL) phase in the larger doping region, we identify two TLL regimes with different features of charge density correlation. At the quarter filling (1/21/21/2 doping level), we find that the strong dimer orders of bond energy in the open system actually decay algebraically and thus do not indicate a spontaneous translational symmetry breaking. Our results show that in the LEL phase of the two-leg ladder, the NNN couplings seem to play the similar role as that on the wider $t\text{\ensuremath{-}}J$ cylinder, and studies on this more accessible system can be helpful toward understanding the emergence of the remarkable ddd-wave superconducting phase on the wider system.
We study the effect of doping away from half-filling in weakly (but finitely) interacting N-leg Hubbard ladders using renormalization-group and bosonization techniques. For a small on-site repulsion U, the N-leg Hubbard ladders are equivalent to an N-band model, where at half-filling the Fermi velocities are ${v}_{1}{=v}_{N}<{v}_{2}{=v}_{N\ensuremath{-}1}<\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}.$ We then obtain a hierarchy of energy scales, where the band pairs $(j,N+1\ensuremath{-}j)$ are successively frozen out. The low-energy Hamiltonian is then the sum of N/2N/2N/2 [or $(N\ensuremath{-}1)/2$ for N odd] two-leg ladder Hamiltonians without gapless excitations (plus a single chain for N odd with one gapless spin mode)---similar to the N-leg Heisenberg spin ladders. The energy scales lead to a hierarchy of gaps. Upon doping away from half-filling, the holes first enter the band(s) with the smallest gap: For odd N, the holes enter first the nonbonding band (N+1)/2(N+1)/2(N+1)/2 and the phase is a Luttinger liquid, while for even N, the holes enter first the band pair (N/2,N/2+1)(N/2,N/2+1)(N/2,N/2+1) and the phase is a Luther-Emery liquid, similar to numerical treatments of the $t\ensuremath{-}J$ model, i.e., at and close to half-filling, the phases of the Hubbard ladders for small and large U are the same. For increasing doping, hole pairs subsequently enter at critical dopings the other band pairs $(j,N+1\ensuremath{-}j)$ (accompanied by a diverging compressibility): The Fermi surface is successively opened by doping, starting near the wave vector $(\ensuremath{\pi}/2,\ensuremath{\pi}/2).$ Explicit calculations are given for the cases N=3N=3N=3 and 4.