Symmetry-protected topological phases in a two-leg $\text{SU}\left(N\right)$ spin ladder with unequal spins

Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective $\mathrm{SU}\left(N\right)$ group (or its subgroup ${\mathbb{Z}}_N\times {\mathbb{Z}}_N)$ with $N>2$. The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $\left(\mathit{N}\right)$ and antifundamental $\left(\overline{\mathit{N}}\right)$ representations of $\mathrm{SU}\left(N\right)$. We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg $\mathrm{SU}\left(N\right)$ spin ladder in which the spins in the first chain transform in $\mathit{N}$ and the second in $\overline{\mathit{N}}$. In particular, we map out the phase diagram for $N=3$ and 4 to show that all the possible symmetry-protected topological phases with projective $\mathrm{SU}\left(N\right)$ symmetry appear in this simple ladder model.

Figure 1

(a) $\mathit{N}\text{−}\overline{\mathit{N}}$ two-leg ladder given by Eq. (1) and (b) its equivalent representation.

Figure 2

(a) Four degenerate states at the leading order in $1/N$ when $J_{\times }>J_{\perp }$. Crossed dimer (CD) and slanted dimer (SD) phases. In CD, translation symmetry is broken spontaneously, while in SD, reflection symmetry is broken. (b) Two chiral SPT states constructed out of “preformed” adjoint spins on individual rungs. In terms of valence-bond covering, these states consist of many dimer-pairing patterns including long-range ones. The SD and chiral SPT phases share the same topological properties.

Figure 3

Phase diagram of $\mathit{N}\text{−}\overline{\mathit{N}}$ ladder (1) (with $J_{\parallel }>0)$ in the large-$N$ limit. On the line $J_{\times }=J_{\perp }$, an effective quantum dimer model (24) emerges at $\text{O}(1/N)$. On the equivalent lattice shown in Fig. 1, CD and SD respectively translate to columnar dimer and staggered dimer. The two degenerate SD states can be interpreted as two chiral SPT states with nontrivial edge states.

Figure 4

Phase diagram for $N=3$ vs $\left(J_{\perp },J_{\times }\right)$ obtained from DMRG simulations on ladders of length $L=240$. Except in the trivial RD phase (which has no order parameter), symbols' sizes are proportional to the values of each order parameter; see text. For completeness, data obtained for the strong-coupling effective Hamiltonian (16) are also plotted and correspond to the limit $J_{\perp }\to -\infty$. Note that the fine-tuned line $J_{\perp }=J_{\times }$ predicted by field theory seems to describe the SPT-RD boundary in the weak-coupling region.

Figure 5

Local quantities obtained by DMRG simulations on ladders of length $L=240$ for $J_{\perp }=-6$ and $J_{\times }=0.5$. The average values of $H_{1,i}^1\approx H_{2,i}^1\approx 0$ (below 0.0001 in absolute values) are not shown. We can see a large difference in the diagonal bond strengths, which is a direct evidence of reflection-symmetry breaking in the chiral SPT phase. By summing up the averages of the diagonal (Cartan) generators around the right edge, we can detect the class-1 SPT phase, with edge state weights $(H_1,H_2)=(0.0,-0.816)\simeq (0,-\sqrt{2/3})$ corresponding to one of the three states in the fundamental representation of SU(3) [precisely, the state with weight ${\vec{\omega }}_3$; see Eq. (E10) for the definition of the weights].

Figure 6

Entanglement spectra obtained by DMRG simulations for a half -system. (a) $N=3$ SPT phase with $J_{\perp }=-6$ and $J_{\times }=0.5$. (b) $N=4$ class-2 SPT phase with $J_{\perp }=-4$ and $J_{\times }=-1$. (c) $N=4$ class-1 SPT phase with $J_{\perp }=-4$ and $J_{\times }=1$. Note that the entanglement spectra are in full agreement with Refs. [44, 58, 59, 87].

Figure 7

Phase diagram for $N=4$ vs $\left(J_{\perp },J_{\times }\right)$ obtained from DMRG simulations on ladders of length $L=128$. Except in the trivial (RD) phase (which has no order parameter), symbols' sizes are proportional to the values of each order parameter; see text. For completeness, data obtained on the strong-coupling effective Hamiltonian (16) are also plotted and correspond to the limit $J_{\perp }\to -\infty$.

Figure 8

Local quantities obtained by DMRG simulations on ladders of length $L=128$ for $J_{\perp }=-4$ and (a) $J_{\times }=0$ or (b) $J_{\times }=0.5$. By summing up the averages of the (diagonal) Cartan generators over the right edge, we can detect, respectively, (a) the class-2 SPT phase, with edge state weights $(H_1,H_2,H_3)=(-0.707,0.408,-0.577)\simeq (-1/\sqrt{2},1/\sqrt{6},-1/\sqrt{3})$ corresponding to one of the states in the self-conjugate antisymmetric $\mathbf{6}$ representation of SU(4); (b) the class-3 SPT phase, with edge state weights $(H_1,H_2,H_3)=(0.707,-0.408,-0.289)\simeq (1/\sqrt{2},-1/\sqrt{6},-1/\sqrt{12})$ corresponding to one of the four states in the $\overline{\mathbf{4}}$ representation of SU(4) precisely, state with weight $-{\vec{\omega }}_2$, see Eq. (E11) and very different diagonal bond amplitudes.