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arXiv:cond-mat/9712143 · Physical Review Letters

Renormalized SO(5) Symmetry in Ladders with Next-Nearest-Neighbor Hopping

Enrico Arrigoni, Werner R. Hanke

Published Mar 8, 1999• 29 citations
arXivHTMLPDF

Abstract

We study the occurrence of SO(5) symmetry in the low-energy sector of two-chain Hubbard-like systems by analyzing the flow of the running couplings ( $g$-ology) under renormalization group in the weak-interaction limit. It is shown that SO(5) is asymptotically restored for low energies for rather general parameters of the bare Hamiltonian. This holds also with the inclusion of a next-nearest-neighbor hopping which explicitly breaks particle-hole symmetry if one accounts for a different single-particle weight for the quasiparticles of the two bands of the system. The physical significance of this renormalization SO(5) symmetry is discussed.

DOI
10.1103/physrevlett.82.2115

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We study the occurrence of SO(5) symmetry in the low-energy sector of two-chain Hubbard-like systems by analyzing the flow of the running couplings ( ggg-ology) under renormalization group in the weak-interaction limit. It is shown that SO(5) is asymptotically restored for low energies for rather general parameters of the bare Hamiltonian. This holds also with the inclusion of a next-nearest-neighbor hopping which explicitly breaks particle-hole symmetry if one accounts for a different single-particle weight for the quasiparticles of the two bands of the system. The physical significance of this renormalization SO(5) symmetry is discussed.
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 ggg-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 ggg-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.
We study, using a perturbative renormalization group technique, the phase diagrams of bond-aligned and diagonal Hubbard ladders defined as sections of a square lattice with nearest-neighbor and next-nearest-neighbor hopping. We find that for not too large hole doping and small next-nearest-neighbor hopping, the bond-aligned systems exhibit a fully spin-gapped phase while the diagonal systems remain gapless. Increasing the next-nearest-neighbor hopping typically leads to a decrease of the gap in the bond-aligned ladders, and to a transition into a gapped phase in the diagonal ladders. Embedding the ladders in an antiferromagnetic environment can lead to a reduction in the extent of the gapped phases. These findings may suggest a relation between the orientation of hole-rich stripes and superconductivity as observed in ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$.
We study a system of two Tomonaga-Luttinger models coupled by a small transverse hopping (a two-chain ladder). We use Abelian and non-Abelian bosonisation to show that the strong coupling regime at low energies can be described by an SO(5)1_11​ WZW model (or equivalently 5 massless Majorana fermions) deformed by symmetry breaking terms that nonetheless leave the theory critical at T=0. The SO(5) currents of the theory comprise the charge and spin currents and linear combinations of the so-called pi operators (S.C. Zhang, Science 275, 1089 (1997)) which are local in terms both of the original fermions and those of the effective theory. Using bosonisation we obtain the asymptotic behaviour of all correlation functions. We find that the 5 component ``superspin'' vector has power law correlations at T=0; other fermion bilinears have exponentially decaying correlations and the corresponding tendencies are suppressed. Conformal field theory also allows us to obtain the energies, quantum numbers, and degeneracies of the low-lying states and fit them into deformed SO(5) multiplets.
We study a system of two Tomonaga-Luttinger models coupled by a small transverse hopping (a two-chain ladder). We use Abelian and non-Abelian bosonization to show that the strong coupling regime at low energies can be described by an SO(5)1\mathrm{SO}(5{)}_{1}SO(5)1​ Wess-Zumino-Witten model (or equivalently five massless Majorana fermions) deformed by symmetry-breaking terms that nonetheless leave the theory critical at T=0.T=0.T=0. The SO(5) currents of the theory comprise the charge and spin currents and linear combinations of the so-called pi operators [S.C. Zhang, Science 275,275,275, 1089 (1997)], which are local in terms both of the original fermions and those of the effective theory. Using bosonization we obtain the asymptotic behavior of all correlation functions. We find that the five-component ``superspin'' vector has power-law correlations at T=0;T=0;T=0; other fermion bilinears have exponentially decaying correlations and the corresponding tendencies are suppressed. Conformal field theory also allows us to obtain the energies, quantum numbers, and degeneracies of the low-lying states and fit them into deformed SO(5) multiplets.
We study the phase diagram of the one-dimensional Hubbard model with next-nearest-neighbor hopping using exact diagonalization, the density-matrix renormalization group, the Edwards variational ansatz, and an adaptation of weak-coupling calculations on the two-chain Hubbard model. We find that a substantial region of the strong-coupling phase diagram is ferromagnetic, and that three physically different limiting cases are connected in one ferromagnetic phase. At a point in the phase diagram at which there are two Fermi points at weak coupling, we study carefully the phase transition from the paramagnetic state to the fully polarized one as a function of the on-site Coulomb repulsion. We present evidence that the transition is second order and determine the critical exponents numerically. In this parameter regime, the system can be described as a Luttinger liquid at weak coupling. We extract the Luttinger-liquid parameters and show how their behavior differs from that of the nearest-neighbor Hubbard model. The general weak-coupling phase diagram can be mapped onto that of the two-chain Hubbard model. We exhibit explicitly the adapted phase diagram and determine its validity by numerically calculating spin and charge gaps using the density-matrix renormalization group.
In this work, we present numerical results which support SO(5) symmetry as a concept unifying superconductivity and antiferromagnetism in the high-temperature superconductors. Using exact cluster diagonalization, we verify that low-energy states of the $t\ensuremath{-}J$ model, a widely used microscopic model for the high-Tc{T}_{c}Tc​ cuprates, form SO(5) symmetry multiplets. Our results show that the ddd-wave superconducting ground states away from half-filling are obtained from the higher-spin states at half-filling through SO(5) rotations. The SO(5) symmetry is found to be robust against the inclusion of longer-ranged Coulomb repulsion and next-nearest-neighbor hopping.