Quantum dimer phases in a frustrated spin ladder: Effective field theory approach and exact diagonalization

The phase diagram of a frustrated $S=1∕2$ antiferromagnetic spin ladder with additional next-nearest neighbor exchanges, both diagonal and in-chain, is studied by a weak-coupling effective field theory approach combined with exact diagonalization for finite systems. In addition to two known phases with rung-singlet and Haldane-type ground states, we observe two new phases with dimerization along the chains. Furthermore, the transitions between the different phases are studied and shown to be either first order or to belong to the universality class of the two-dimensional Ising model. The nature of elementary excitations is discussed briefly.

Figure 1

Structure of the spin ladder with next-nearest-neighbor interactions.

Figure 2

Typical configuration in the RVB state of the antiferromagnetic ladder.

Figure 3

Topology of the phase diagram as captured within the weak-coupling approach. Phases are as follows. A, rung singlet, B, Haldane, C, columnar dimer, and D, staggered dimer. The dashed line represents a first-order phase transition line, whereas continuous lines stand for Ising transition lines. Deviations of the dashed line at weak coupling from the straight line $J_{\perp }=2J_{\times }$ and enlargement of the columnar dimer phase are due to second-order processes as found in Ref. 6.

Figure 4

(Color online) Numerical results for the lowest excited states with $S=0$ and $k=\pi$ at $J_2=J∕2$, $J_{\perp }=0.2J$ as a function of $J_{\times }$. Lines are for $N=16$ rungs; symbols show extrapolations to the thermodynamic limit. The big filled triangles denote estimates for the locations of the different phase transitions. For details see the text.

Figure 5

(Color online) Numerical results for the phase diagram in the $J_2=J∕2$ plane. Rung-singlet, Haldane, columnar dimer, and staggered dimer phases can be distinguished. Crosses have been determined by the phenomenological renormalization group condition (25) with $N_1=14$, $N_2=16$; the corresponding lines with $N_1=12$, $N_2=14$. For further details and the nature of the phase transitions see the text.

Figure 6

(Color online) Scaled energies of the $k=\pi$, $k_y=0$ excited level for $J_{\perp }=0.9J$, $J_2=J∕2$ as a function of $J_{\times }$. Lines connect actual numerical results, which are shown by symbols. According to the phenomenological renormalization group condition (25), crossings of levels for different $N$ are estimates for the boundaries of the columnar dimer phase.

Figure 7

(Color online) The lowest excited levels with total spin $S=0$, 1, and 2 on a ladder with $N=12$ rungs for $J_{\times }=0$, $J_2=J∕2$ as a function of $J_{\perp }$ on a semilogarithmic scale.