Staggered orbital currents in the half - filled two - leg ladder

One of the most intriguing phases of strongly correlated electrons is known variously as the ‘‘orbital antiferromagnet,’’ the ‘‘staggered flux phase,’’ or the ‘‘d-density wave.’’ It is characterized by circulating currents which produce local magnetic moments aligned in an antiferromagnetic ~staggered! way. As a consequence, timereversal symmetry as well as translational and rotational symmetries are spontaneously broken. Another phase, the ‘‘circulating current phase,’’ is somewhat similar, but does not break translational symmetry. These phases have received considerable attention lately, due to their possible relevance to the pseudogap region in the phase diagram of the cuprates. A recent neutron scattering experiment on underdoped YBa2Cu3O6.6 has been interpreted 14 as evidence for these staggered orbital currents. In this paper we focus on the half-filled two-leg ladder, which is the simplest system that can support the staggered flux ~SF! phase ~see Fig. 1!. As the order parameter of this phase breaks a discrete (Z2) symmetry, the possibility of true long-range order ~LRO! of the currents is not a priori excluded at zero temperature in this one-dimensional system, in contrast to the situation for order parameters that break continuous symmetries, which causes their correlations to exhibit at most quasi-LRO with power-law decay. For weak interactions, the ladder can be treated using bosonization and the perturbative renormalization group ~RG!. For this case, the SF phase has been found in the phase diagram for spinless electrons at half-filling. Furthermore, away from half-filling, regions with dominant tendencies toward SF ordering have been found both for spinless and spinful electrons. Note that for general ~i.e., incommensurate! fillings, true LRO of the currents is not possible due to the absence of Umklapp interactions ~see Sec. V for a more detailed discussion!. The results for the doped ladder with and without spin were summarized in Ref. 18, which also investigated the effects of disorder. Here we are concerned with the SF phase for spinful electrons in a weakly interacting half-filled two-leg ladder. In contrast to the other weak-interaction studies mentioned so far, in the approach used here the nearest-neighbor hopping parameters t' and t along the rungs and legs, respectively, can be of the same order. We reanalyze the nature of a specific phase found in Ref. 19, and demonstrate that this phase, previously identified to be of spin-Peierls ~SP! type, actually exhibits staggered orbital currents with no dimerization, and