Ground-state phases of a rung-alternated $\text{spin}-\frac{1}{2}$ Heisenberg ladder

The ground-state phase diagram of a Heisenberg $\text{spin}-\frac{1}{2}$ system on a two-leg ladder with rung alternation is studied by combining analytical approaches with numerical simulations. For the case of ferromagnetic leg exchanges a unique ferrimagnetic ground state emerges, whereas for the case of antiferromagnetic leg exchanges several different ground states are stabilized depending on the ratio between exchanges along legs and rungs. For the more general case of a honeycomb-ladder model for the case of ferromagnetic leg exchanges besides the usual rung-singlet and saturated ferromagnetic states we obtain a ferrimagnetic Luttinger liquid phase with both linear and quadratic low-energy dispersions and ground-state magnetization continuously changing with system parameters. For the case of antiferromagnetic exchanges along legs, different dimerized states including states with additional topological order are suggested to be realized.

Figure 1

Geometry of the two-leg ladder with alternated rung exchanges. Antiferromagnetic couplings are chosen along the even rungs, $J_{\perp }>0$.

Figure 2

Analytically conjectured ground-state phase diagram of rung-alternated Heisenberg ladder as function of $J$. $H_1$ denotes a Haldane phase, however only half of the rungs (odd numbered rungs) provide the effective $S=1$ spins, whereas the other half (even numbered rungs) are in the approximate rung-singlet states and mediate antiferromagnetic exchange among effective $S=1$ spins.

Figure 3

Geometry of honeycomb-ladder model. For $\alpha =0$ the rung-alternated ladder presented in Fig. 1 is recovered.

Figure 4

Single-particle dispersion relations (a) deep in ferro state, (b) at the boundary of ferro and ferri states, (c) inside ferri state, (d) at the boundary of ferri and half-ferro states, and (e) inside the half-ferro state. The dashed line indicates the chemical potential.

Figure 5

Effective model for spins belonging to odd rungs valid for $\alpha \simeq 1$ and $J\ll 1$, where spins belonging to even rungs form approximate rung-singlet states. Phase transitions are expected for $J_r<0$.

Figure 6

Cartoon of one of the possible ground-state configurations of the Haldane-dimer phase that can be realized for $\alpha \simeq -1$, $0<J\ll 1$. Spins encircled by open rectangles form approximate rung-singlet state, whereas those encircled by shaded rectangles form rung-triplet states. Six spins encircled by the dotted rectangle form effective $S=1$ spins which are connected via intermediate singlets to produce an effective Haldane chain. In the mixed-diamond chain the singlets depicted above are exact eigenstates and they do not mediate any exchange among the effective $S=1$ spins.

Figure 7

Fidelity susceptibility of the rung-alternated ladder with antiferromagnetic legs for periodic boundary conditions and three different system sizes. In DMRG calculations periodic boundary conditions are assumed (restricting considerably the available ladder lengths) to avoid degeneracies of the Haldane-like states due to edge spins for the open boundaries.

Figure 8

DMRG results for the lowest energy levels in different total $S^z$ subspaces relative to the lowest energy in the $S^z=0$ subspace as function of $\alpha$ for $J=-1$ and $L=96$ rungs for open boundary conditions. The dashed line indicates the boundary of the ferromagnetic state.

Figure 9

Lanczos results of the lowest excited states relative to the ground state obtained for $L=12$ rungs of the effective model Eq. (12) for (a) $J=0.1$ and (b) $J=0.2$. A similar picture is expected to hold for the Honeycomb-ladder model with $L=24$ rungs for the same extent of $\alpha$ and $J$. Periodic boundary conditions are used that allows us to assign a definite lattice momentum to each level.

Figure 10

Numerical ground-state phase diagram of the honeycomb-ladder model in the parameter plane $(J,\alpha )$. NNND-dimer stands for next-nearest-neighbor diagonal dimer phase where dimers are formed along next-nearest-neighbor diagonals involving spins of odd rungs. In the vicinity of $\alpha =-1$ and for $0<J\ll 1$ before the transition from Haldane-dimer to Haldane phase additional phases may occur (e.g., with $M=5$ and $M=7$ as discussed in previous sections). In dimer phases ground states are doubly degenerate in the thermodynamic limit.

Figure 11

Ground-state fidelity susceptibility per site as function of $\alpha$ for (a) $J=0.4$ and for $J=1$ showing two peaks. In DMRG calculations periodic boundary conditions are assumed to avoid degeneracies of the Haldane-like states due to edge spins for open boundaries.

Figure 12

Gap between the ground state and the first excited (triplet) state as function of $\alpha$ for $J=5$ obtained by DMRG using open boundary conditions.