Exact Ground States with Deconfined Gapless Excitations for the Three-Leg Spin-$1/2$ Tube

We consider a spin-$1/2$ tube (a three-leg ladder with periodic boundary conditions) with a Hamiltonian given by two projection operators—one on the triangles and the other on the square plaquettes on the side of the tube—that can be written in terms of Heisenberg and four-spin ring exchange interactions. We identify 3 phases: (i) for strongly antiferromagnetic exchange on the triangles, an exact ground state with a gapped spectrum can be given as an alternation of spin and chirality singlet bonds between nearest triangles; (ii) for ferromagnetic exchange on the triangles, we recover the phase of the spin-$3/2$ Heisenberg chain; (iii) between these two phases, a gapless incommensurate phase exists. We construct an exact ground state with two deconfined domain walls and a gapless excitation spectrum at the quantum phase transition point between the incommensurate and dimerized phases.

Figure 1

The spin tube with a snapshot of the valence bond covering from the exact spin-chirality dimerized ground state. The shaded plaquettes are not satisfied in this particular configuration, and only quantum resonance with other configurations will make all the plaquettes satisfied.

Figure 2

(a) Energy spectrum from exact diagonalization of a tube with 10 triangles and $K_{\Delta }=0$ as a function of momentum along the tube. The empty (filled) symbols denote spin-singlet (triplet) states classified according to the irreducible representations (IR) of $D_3$. (b) Low energy two domain wall excitations in the thermodynamic limit compared to ED spectra of small clusters with symmetry compatible with the variational solution. The thick line below the (shaded) continuum is the bound state.

Figure 3

(a) Schematic representation of one of the two ground states of $\mathcal{H}$ with alternating spin and chirality valence bonds (a translation by a lattice vector gives the other one). The small dots inside the circle represent the individual spins of a triangle; thick solid lines denote valence bonds. The arcs connecting two circles stand for a chirality singlet bond $|rl⟩-|lr⟩$. In (b) and (c) we show the relevant domain walls in odd length tubes: (b) $|{\xi }_i^{\uparrow }⟩$, a triangle with $\tilde{S}=3/2$ (hexagon); (c) the ellipse containing three spin-$1/2$ triangles with chiralities $|rrr⟩+|lll⟩$ and centered at $i+1$ denotes $|{\eta }_{i+1}^{\uparrow }⟩$. Both $|{\xi }^{\uparrow }⟩$ and $|{\eta }^{\uparrow }⟩$ have spin-$1/2$ degrees of freedom.

Figure 4

Relevant two domain wall configurations in the spin singlet sector for $L$ even. In (a) the small ellipse denotes a chirality triplet ($|rl⟩+|lr⟩$) that breaks up into two domain walls. (c), (e), and (f) are generally given as $|{\xi }_i^{\uparrow }{\eta }_j^{\downarrow }-{\xi }_i^{\downarrow }{\eta }_j^{\uparrow }⟩$, $|{\eta }_i^{\uparrow }{\eta }_j^{\downarrow }-{\eta }_i^{\downarrow }{\eta }_j^{\uparrow }⟩$, and $|{\xi }_i^{\uparrow }{\xi }_j^{\downarrow }-{\xi }_i^{\downarrow }{\xi }_j^{\uparrow }⟩$, respectively, where the domain walls can be arbitrarily separated. (b) and (d) show overlapping domain walls—(b) is actually $|{\xi }_i^{\uparrow }{\xi }_{i+1}^{\downarrow }-{\xi }_i^{\downarrow }{\xi }_{i+1}^{\uparrow }⟩$, and in (d) the chirality configuration is $|llrr⟩-|rrll⟩$. Arrows connect states between which the Hamiltonian ${\mathcal{H}}_{K_{\Delta }=0}$ has a nonzero matrix element, the position of the arrows corresponds to the position of $R$ relevant in the overlap.

Figure 5

Schematic phase diagram of the model.