Orbital multicriticality in spin-gapped quasi-one-dimensional antiferromagnets

Motivated by the quasi-one-dimensional antiferromagnet CaV${}_2$O${}_4$, we explore spin-orbital systems in which the spin modes are gapped but orbitals are near a macroscopically degenerate classical transition. Within a simplified model we show that gapless orbital liquid phases possessing power-law correlations may occur without the strict condition of a continuous orbital symmetry. For the model proposed for CaV${}_2$O${}_4$, we find that an orbital phase with coexisting order parameters emerges from a multicritical point. The effective orbital model consists of zigzag-coupled transverse field Ising chains. The corresponding global phase diagram is constructed using field theory methods and analyzed near the multicritical point with the aid of an exact solution of a zigzag $\mathit{XXZ}$ model.

Figure 1

(a) Phase diagram of effective orbital model (1) near the multicritical point at $h=\Delta$, $\lambda =0$. Here $\kappa \propto {\lambda }^2/{\Delta }_s$. The properties of the four orbital phases are described in Table . (b) Schematic global phase diagram of the ANNNI model given in Eq. (3) with $\ell =1$.

Figure 2

Schematic representation of the exactly solvable $U(1)\otimes U(1)$ symmetric zigzag model with $\Delta /\kappa \to \infty$, corresponding to spinless fermions hopping on either even or odd sublattices, with infinite repulsion along zigzag bonds.

Figure 3

(a) Example of a configuration with $N_d=2$ domains in the original zigzag chain, with $n_1=2$ fermions in the first domain at positions $r_{1,1}=2$, $r_{1,2}=6$, and with $n_2=1$ fermions in the second domain located at position $r_{2,1}=9$. (b) At the fictitious lattice with length $L^{\prime }=4$, constructed by keeping only the sites inside dashed rectangles in (a), the occupied sites are $r_1^{\prime }=1$, $r_2^{\prime }=3$, and $r_3^{\prime }=4$. In the original lattice there are only two possible hopping processes, denoted by arcs without crosses, compatible with the infinite repulsion along zigzag bonds. The interaction-forbidden hopping processes are blocked in the fictitious lattice due to the Pauli principle.

Figure 4

(a) Phase diagram of the $U(1)\otimes U(1)$ symmetric model $H_{\mathit{XXZ},2}$ in Eq. (5). The symmetry properties of the phases are given in Table . (b) Adding an infinitesimal $\epsilon$ perturbation in $H_{\mathrm{eff},2}$ in Eq. (5) leads to the opening of an energy gap in the OLz phase of (a); the resulting phase AFOyz has coexisting order parameters as specified in Table .

Figure 5

Schematic global phase diagram of model (3) with $\ell =2$, namely, with NNN transverse couplings. All phases are gapped and characterized according to the broken symmetries as given in Table . Starting from the disordered PO phase, both Ising order parameters ${\eta }_0^y$ and ${\eta }_1^y$ acquire finite values upon crossing line $\mathit{OA}$. In the AFOyz phase the ${\eta }^z$ order parameter is finite, as well as one of the two ${\eta }_m^y$ ($m=0,1$) order parameters. All three critical lines emanating from point $A$ are described by a LL theory, which follows from a formal analogy between the effective theory of point $A$, Eq. (25), and the effective theory of the XYZ spin chain Eq. (27). Point $A$ has an emergent SU(2) symmetry, corresponding to the Heisenberg point of the XYZ spin chain, and realized in terms of the dual $\mu$ variables defined in Eq. (28).