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 $\mathrm{SO}(5{)}_{1}$ Wess-Zumino-Witten model (or equivalently five massless Majorana fermions) deformed by symmetry-breaking terms that nonetheless leave the theory critical at $T=0.$ 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 bosonization we obtain the asymptotic behavior of all correlation functions. We find that the five-component ``superspin'' vector has power-law correlations at $T=0;$ 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 Wess-Zumino-Witten model (or equivalently five 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 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; 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 bosonisation to show that the strong coupling regime at low energies can be described by an SO(5)1 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 consider a system of interacting spinless fermions on a two-leg triangular ladder with $\ensuremath{\pi}/2$ magnetic flux per triangular plaquette. Microscopically, the system exhibits a U(1) symmetry corresponding to the conservation of total fermionic charge and a discrete Z2 symmetry---a product of parity transformation and chain permutation. Using bosonization, we show that, in the low-energy limit, the system is described by the quantum double-frequency sine-Gordon model. On the basis of this correspondence, a rich phase diagram of the system is obtained. It includes trivial and topological band insulators for weak interactions, separated by a Gaussian critical line, whereas at larger interactions a strongly correlated phase with spontaneously broken Z2 symmetry sets in, exhibiting a net charge imbalance and nonzero total current. At the intersection of the three phases, the system features a critical point with an emergent SU(2) symmetry. This non-Abelian symmetry, absent in the microscopic description, is realized at low energies as a combined effect of the magnetic flux, frustration, and many-body correlations. The criticality belongs to the SU(2)1 Wess-Zumino-Novikov-Witten universality class. The critical point bifurcates into two Ising critical lines that separate the band insulators from the strong-coupling symmetry broken phase. We establish an analytical connection between the low-energy description of our model around the critical bifurcation point on one hand and the Ashkin-Teller model and a weakly dimerized XXZ spin-1/2 chain on the other. We complement our field-theory understanding via tensor network simulations, providing compelling quantitative evidences of all bosonization predictions. Our findings are of interest to up-to-date cold atom experiments utilizing Rydberg dressing that have already demonstrated correlated ladder dynamics.
A Parity Signature Distinguishing SO(5)1 from Spin(4)1 Boundaries
The level-one Wess-Zumino-Witten models SO(5)1 and Spin(4)1 share the same total quantum dimension and therefore the same topological entanglement entropy, yet they differ in the parity of their Majorana channel count: five (odd) versus four (even). We show that this even/odd difference is a sharp, falsifiable boundary signature even though the standard entropic diagnostic is blind to it. Under non-abelian bosonization SO(5)1 is five Majorana fermions, and the odd count leaves one mode unpaired and the spinor primary a single Ising-class sector of quantum dimension 2, where the even neighbor Spin(4)1 pairs cleanly and its spinor splits into two abelian sectors. On an explicit coupled-Majorana-chain realization, a class-D Z2 index and a two-parameter phase scan show that a protected unpaired-Majorana parity persists across an entire gapped phase for the odd (n=5) case and is absent for the even (n=4) control. On the interacting bilinear-biquadratic chains at the Reshetikhin point, a charge-conserving entanglement-spectrum computation resolves the two cases directly: the SO(5)1 tower carries the five-fold vector multiplet including the zero-weight state, the Spin(4)1 tower a four-fold shell without it, and the topological entanglement entropy γ=log2 is identical for both and cannot separate them where the parity and channel count can. A direct diagonalization of the genuine twisted sector resolves the spinor multiplet, which appears as the tensor square 4⊗4=1⊕5⊕10 rather than the bare chiral spinor, the expected consequence of the Nielsen-Ninomiya obstruction to a single chiral mode on a non-chiral lattice. We also propose a candidate mechanism in which the open non-abelian sector is completed non-locally through a three-dimensional Chern-Simons bulk; the demonstrated lattice results stand independently of this conjecture.
We discuss properties of a recently proposed exactly SO(5)-symmetric ladder model. In the strong coupling limit we demonstrate how the SO(3)-symmetric description of spin ladders in terms of bond bosons can be upgraded to an SO(5)-symmetric bond-boson model, which provides a particularly simple example for the concept of SO(5) symmetry. Based on this representation we show that antiferromagnetism and superconductivity can be understood as condensation of either magnetic or charged bosons into a resonating-valence-bond vacuum. We identify exact eigenstates of a finite cluster with general multiplets of the SO(5) group, and present numerical results for the single particle, spin and charge spectra of the SO(5)-symmetric model and identify ``fingerprints'' of SO(5) symmetry in these. In particular we show that SO(5) symmetry implies a ``generalized rigid band behavior'' of the photoemission spectrum, i.e., spectra for the doped case are rigorously identical to spectra for spin-polarized states at half filling.
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.