Competing orders and hidden duality symmetries in two-leg spin ladder systems

A unifying approach to competing quantum orders in generalized two-leg spin ladders is presented. Hidden relationship and quantum phase transitions among the competing orders are thoroughly discussed by means of a low-energy field theory starting from an SU(4) quantum multicritical point. Our approach reveals that the system has a relatively simple phase structure in spite of its complicated interactions. On top of the U(1) symmetry which is known from previous studies to mix up antiferromagnetic order parameter with that of the $p$-type nematic, we find an emergent U(1) symmetry which mixes order parameters dual to the above. On the basis of the field-theoretical and variational analysis, we give a qualitative picture for the global structure of the phase diagram. Interesting connection to other models (e.g., the bosonic $t\text{−}J$ model) is also discussed.

Figure 1

(a) Two-leg spin ladder. The unit of constructing the model Hamiltonian (see Sec. 2) is a pair of spin-$1∕2$’s enclosed in a broken line (called rung in the text). Any site can be labeled by the chain index (1,2) and the rung index $r$. (b) and (c): Two typical states appearing in two-leg ladders. The usual rung-singlet phase (b) and the staggered-dimer phase (c) to be discussed in the text. Arrows denote spin-singlet pairs.

Figure 2

(Color online) Changes in the symmetry as the parameters are varied. Our starting point denoted by “SU(4)” corresponds to the choice $A=B=C=D$, $E=F=G=0$. “Duality” $\mathcal{D}$ maps a model with $B>D$ onto another with $B<D$ and vice versa. Note that the full symmetry for generic cases (including the spin-orbital case) is $\mathrm{SU}{\left(2\right)}_{\text{spin}}\times {\mathbb{Z}}_2$.

Figure 3

(Color online) RG flow for the SO and dSO models. The origin [shown as SU(4) FP] corresponds to the level-1 SU(4) WZW model and a line $g_1=g_3$ to self-dual models [SU(4) symmetric, in this case]. A dashed line roughly corresponds to a path across the SU(4) Sutherland model [extended $\text{massless}\to \mathrm{SU}\left(4\right)\to \text{staggered}$ dimer (SD)] discussed in literature (e.g., Ref. 57). An asymptotic trajectory in the lower-right portion (highlighted by a thick arrow) is responsible for the spontaneous breakdown of translational (for SO model) and time-reversal (for dSO model) symmetries.

Figure 4

(Color online) Relationship among the four dominant phases. Ratio of the couplings $g_1$, $g_2$, $g_3$, and $g_4$ along each symmetric ray is shown.

Figure 5

(Color online) Schematic picture illustrating how the doublet of order parameters is fixed: (a) At the SU(4) point, the doublet fluctuate both in the radial and angular directions. (b) For the self-dual models, the modulus of it is fixed by the interactions in the SU(3) sector, whereas fluctuations in the azimuthal direction are still gapless. (c) The remaining angular fluctuation gets locked by a small deviation from the self-dual models.

Figure 6

(Color online) A schematic picture illustrating the effects of ${\mathcal{H}}_6$ and the anisotropy $\epsilon =B-D$ in the spin-chirality plane on the global structure of the phase diagram of our model. According to the sign of the anisotropy, we have two different orderings; an anisotropic SF in the $x$ direction ($\mathcal{T}$ even) for $\epsilon >0$ and one in the $y$ direction ($\mathcal{T}$ odd) for $\epsilon <0$. The duality symmetry $\mathcal{D}$ interchanges the region with $\epsilon >0$ and that with $\epsilon <0$. On both sides of the region enclosed by a dashed line (dominant singlet fluctuations), we find conventional phases (spin-1 or rung singlet), which can be interpreted as insulating ones.

Figure 7

(Color online) (a) Variational phase diagram obtained by the ansatz (117). Shown is the pseudospin direction $(\varphi ,\theta )$ for each phase: conventional rung-singlet and rung-triplet (Haldane or VBS) phases are spin-polarized states while the SD and SC phases may be viewed as canted states with different $XY$ projections. (b) Typical spin-singlet configuration contained in our variational wave function (117). Ovals and open circles respectively denote triplet (occupied sites) and singlet (unoccupied sites) rungs. The spin-1 bosons totally form a spin-singlet valence-bond state. Due to motion and pair creation or annihilation of triplet bosons, the wave function (117) describes a strongly fluctuating state.

Figure 8

(Color online) (a) A two-leg ladder as a weakly coupled plaquettes. Strongly coupled plaquettes, on which pseudospin $1∕2$’s and coarse-grained spin-1’s are defined, are shown by thick lines. All interactions (both two spin and four spin) connecting these plaquettes are multiplied by $\lambda (⪡1)$. (b), (c) Two triplet states which constitute a (nearly) degenerate ground-state manifold in the vicinity of the SU(4) point. Two spin-$1∕2$’s contained in each oval form a spin triplet. In both (b) and (c), two triplets (ovals) are antisymmetrized to form a total spin triplet on a plaquette.