Symmetry-protected topological phases and competing orders
in a spin-$\frac{1}{2}$ XXZ ladder with a four-spin interaction

The concept of symmetry-protected topological (SPT) phases has been proposed and fruitfully developed over the last decade [1, 2, 3, 4, 5, 6]. A SPT phase is a featureless gapped phase that does not exhibit spontaneous symmetry breaking but is distinguished from a trivial phase (i.e. a phase that includes a site-factorized product state) as long as certain symmetries are imposed. For example, Haldane phases [7, 8, 9] of odd-integer-spin chains are distinguished from trivial phases in the presence of the discrete spin rotation symmetry $D_{2}(=\mathbb{Z}_{2}\times\mathbb{Z}_{2})$, time-reversal symmetry, or bond-centered inversion symmetry [1, 10, 11, 12, 13]. The Haldane phases with odd integer spins have been characterized by a string order parameter [14], hidden symmetry breaking [15, 16], a twist operator [17, 18], degeneracy in the edge states [19], the entanglement spectrum [10], and topological indices [10, 20, 21, 22]. Classification of SPT phases of bosons has been achieved by using the projective representations of the symmetry group in one dimension [21, 22, 23] and group cohomology theory in higher dimensions [24, 25].

A simple extension of the spin-$1$ Haldane phase can be found in a spin-$\frac{1}{2}$ two-leg ladder. Consider a spin-$\frac{1}{2}$ ladder with Heisenberg interactions $J$ and $J_{\perp}$ along the legs and rungs, respectively, where $J$ is assumed to be antiferromagnetic; see Fig. 1. Field-theoretical analyses indicate that the presence of the rung interaction $J_{\perp}\neq 0$ immediately leads to gapped phases whose properties depend on the sign of $J_{\perp}$ [26, 27, 28]. In the gapped phase for ferromagnetic $J_{\perp}<0$, effective spin-$1$ degrees of freedom emerge on the rungs and collectively form the Haldane state. This state can equivalently be viewed as a superposition of various singlet-covering states on a ladder. In contrast, the gapped phase for antiferromagnetic $J_{\perp}>0$ can be understood from the limit $J_{\perp}\to\infty$, where the ground state is a product state of singlet pairs on the rungs, known as the rung singlet (RS) state. The gapped phases for $J_{\perp}<0$ and $J_{\perp}>0$ are thus called the Haldane and RS phases, respectively. These phases are distinguished in terms of two types of string order parameters [29, 30, 31, 32] [shown in Eq. (5) later].

Liu et al. [33] have conducted a detailed classification of SPT phases in a spin-$\frac{1}{2}$ two-leg ladder for the symmetry group $D_{2}\times\sigma$, where $\sigma$ is the symmetry with respect to the interchange of the two legs. They have found three new SPT phases termed $t_{\mu}~{}(\mu=x,y,z)$, all of which have symmetry-protected two-fold degenerate edge states on each end of an open ladder, with unique responses to magnetic fields. For example, the edge states in the $t_{z}$ phase are split by the magnetic field along the $z$ direction, but not by the fields in the $x$ and $y$ directions. The $t_{\mu}$ phase is related with the Haldane phase (termed $t_{0}$ in Ref. [33]) under the $\pi$ rotation of spins about the $\mu$-axis on the first leg, $U_{1}^{\mu}(\pi):=\exp\quantity(i\pi\sum_{j}S_{1,j}^{\mu})$, which provides some intuition on the ground state. Specifically, a singlet pair between spins on different legs turns into a twisted singlet $\ket{1,z}:=\left(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow}\right)/\sqrt{2}$ under $U_{1}^{z}(\pi)$. It has also been argued that the trivial phase (i.e., the phase that includes rung-factorized states) is divided into four phases, termed rung-$\ket{0,0}$ and rung-$\ket{1,\mu}~{}(\mu=x,y,z)$, if the translational symmetry is further imposed. The rung-$\ket{0,0}$ phase corresponds to the conventional RS phase, and the rung-$\ket{1,\mu}$ phase is obtained by operating the unitary transformation $U_{1}^{\mu}(\pi)$ on the rung-$\ket{0,0}$ phase. A highly anisotropic XYZ model on a ladder has been studied, which exhibits competition among the $t_{0}$, $t_{z}$, rung-$\ket{0,0}$, and rung-$\ket{1,z}$ phases and a variety of magnetic phases (see also Ref. [34] and Appendix A). Fuji has given a field-theoretical description of a variety of SPT and symmetry-broken phases of a spin-$\frac{1}{2}$ XXZ ladder [35, 36]. Following him, we henceforth use the terms, the Haldane* and RS* phases, to refer to the $t_{z}$ and rung-$\ket{1,z}$ phases, respectively, where a star indicates the operation of $U_{1}^{z}(\pi)$ or equivalently the twist of singlet pairs between the legs. As the study of these phases has so far been limited in literature, it is worthwhile to further investigate when these phases emerge and how they compete with other phases in concrete spin models.

In this paper, we study a simple extension of the spin-$\frac{1}{2}$ Heisenberg ladder that has XXZ anisotropy $\Delta$ and a four-spin interaction $J_{4}$ (Fig. 1). The Hamiltonian of our model is given by

	$\displaystyle H=$	$\displaystyle J\sum_{\alpha=1,2}\sum_{j}(\bm{S}_{\alpha,j}\cdot\bm{S}_{\alpha,j+1})_{\Delta}+J_{\perp}\sum_{j}(\bm{S}_{1,j}\cdot\bm{S}_{2,j})_{\Delta}$
		$\displaystyle+J_{4}\sum_{j}(\bm{S}_{1,j}\cdot\bm{S}_{1,j+1})(\bm{S}_{2,j}\cdot\bm{S}_{2,j+1}),$		(1)

where

$$(\bm{S}_{\alpha,i}\cdot\bm{S}_{\beta,j})_{\Delta}:=S^{x}_{\alpha,i}S^{x}_{\beta,j}+S^{y}_{\alpha,i}S^{y}_{\beta,j}+\Delta S^{z}_{\alpha,i}S^{z}_{\beta,j}.$$

(2)

The four-spin interaction with $J_{4}>0$ ($J_{4}<0$) introduces effective repulsion (attraction) between singlet pairs on the upper and lower edges of each plaquette. Throughout this paper, we take the leg interaction $J=1$ as the unit of energy. In contrast to the highly anisotropic models studied in Refs. [33, 34, 35], we investigate a regime around the isotropic case $\Delta=1$. By means of effective field theory for $|J_{\perp}|,|J_{4}|\ll 1$, we obtain rich ground-state phase diagrams that consist of eight distinct gapped phases, as schematically shown in Figs. 2 and 3. We also perform numerical calculations based on the variational uniform matrix product state (VUMPS) algorithm [37, 38] for $J_{\perp}=\pm 1$ to demonstrate that essentially the same phase structures continue for $|J_{\perp}|,|J_{4}|=O(1)$, as shown in Figs. 4 and 10 later. A notable feature as compared to Refs. [33, 34] is that the four featureless phases, i.e., the RS, RS*, Haldane, and Haldane* phases compete not only with magnetic phases but also with staggered dimer (SD) and columnar dimer (CD) phases, which have valence bond solid (VBS) long-range orders breaking the translational symmetry. It is also remarkable that the RS* and Haldane* phases, which have highly anisotropic wave functions involving twisted singlet pairs, appear even in the vicinity of the isotropic case $\Delta=1$. Below we briefly review previous studies on the model (I) and highlight our contributions.

The model (I) has been studied intensively in the isotropic case $\Delta=1$ [39, 40, 41, 42, 43, 44, 45], which corresponds to the horizontal axes of Figs. 2 and 3. The SD and CD phases can be interpreted as a consequence of effective repulsion or attraction between singlet pairs due to $J_{4}$. These phases have two-fold degenerate ground states below an excitation gap and are characterized by the order parameters

	$\displaystyle\expectationvalue{\mathcal{O}_{\text{SD}}(j)}$	$\displaystyle=\frac{1}{4}\langle\bm{S}_{1,j-1}\cdot\bm{S}_{1,j}-\bm{S}_{2,j-1}\cdot\bm{S}_{2,j}$
		$\displaystyle~{}~{}~{}~{}~{}~{}-\bm{S}_{1,j}\cdot\bm{S}_{1,j+1}+\bm{S}_{2,j}\cdot\bm{S}_{2,j+1}\rangle,$		(3a)
	$\displaystyle\expectationvalue{\mathcal{O}_{\text{CD}}(j)}$	$\displaystyle=\frac{1}{4}\langle\bm{S}_{1,j-1}\cdot\bm{S}_{1,j}+\bm{S}_{2,j-1}\cdot\bm{S}_{2,j}$
		$\displaystyle~{}~{}~{}~{}~{}~{}-\bm{S}_{1,j}\cdot\bm{S}_{1,j+1}-\bm{S}_{2,j}\cdot\bm{S}_{2,j+1}\rangle.$		(3b)

The translational symmetry as well as a $Z_{2}$ symmetry with respect to the rung-centered reflection (${\bm{S}}_{\alpha,j}\mapsto{\bm{S}}_{\alpha,-j}$) are spontaneously broken in these phases. Field-theoretical analyses for weak inter-chain couplings ($|J_{\perp}|,|J_{4}|\ll 1$) [39, 46, 44, 45] suggest that the RS-SD and Haldane-CD transitions are continuous and described by the SU$(2)_{2}$ Wess-Zumino-Witten (WZW) theory with the central charge $c=3/2$, although a possibility of a first-order transition is not excluded. In this approach, the transition points are estimated to be $J_{4}\approx 2.05J_{\perp}$ for both signs of $J_{\perp}$ for $|J_{\perp}|,|J_{4}|\ll 1$ [44]. More detailed phase diagrams on the $J_{\perp}$-$J_{4}$ plane have been obtained numerically [42, 43, 45]. In particular, the exact diagonalization result of Ref. [42] for the RS-SD transition is consistent with the $c=3/2$ criticality for $0.5\lesssim J_{\perp}\lesssim 1.5$.

We have started our analysis of the anisotropic case $\Delta\neq 1$ in Ref. [47], and we extend it substantially in the present work. The obtained phase diagrams in Figs. 2 and 3 look like mirror images of each other ¹¹1 A useful way to relate the phases in Figs. 2 and 3 is to shift the lower leg of the ladder by one lattice spacing. In the bosonization formalism, this can be expressed as the shifts of $2\sqrt{\pi}\phi_{2}$ and $\sqrt{\pi}\theta_{2}$ by $-\pi$ and $\pi$, respectively, as seen in Eq. (7). This corresponds to the shifts of $2\sqrt{2\pi}\phi_{\pm}$ and $\sqrt{2\pi}\theta_{-}$ by $\mp\pi$ and $-\pi$, respectively. The reason for this qualitative relation between Figs. 2 and 3 is as follows. In the bosonized expressions of spin and dimer operators in Eq. (7), the staggered components play a leading role. Therefore, an antiferromagnetic interaction on a rung, $J_{\perp}(\bm{S}_{1,j}\cdot\bm{S}_{2,j})_{\Delta}$ with $J_{\perp}>0$, is qualitatively similar to a ferromagnetic interaction in a diagonal direction, $-J_{\perp}(\bm{S}_{1,j}\cdot\bm{S}_{2,j-1})_{\Delta}$, which is then transformed to a ferromagnetic interaction on a rung after the shift of the lower leg. A similar transformation can also be performed for $J_{4}$. . The Haldane* and RS* phases, which are both featureless, appear for easy-plane anisotropy $\Delta<1$. The Néel and stripe Néel (SN) phases, which have magnetic long-range orders along the $z$-axis, appear for easy-axis anisotropy $\Delta>1$. The Néel and SN phases have two-fold degenerate ground states below an excitation gap, and are characterized by the order parameters

	$\displaystyle\expectationvalue{\mathcal{O}_{\text{N\'{e}el}}(j)}$	$\displaystyle=\frac{1}{4}\expectationvalue{S^{z}_{1,j}-S^{z}_{2,j}-S^{z}_{1,j+1}+S^{z}_{2,j+1}},$		(4a)
	$\displaystyle\expectationvalue{\mathcal{O}_{\text{SN}}(j)}$	$\displaystyle=\frac{1}{4}\expectationvalue{S^{z}_{1,j}+S^{z}_{2,j}-S^{z}_{1,j+1}-S^{z}_{2,j+1}}.$		(4b)

A $Z_{2}$ symmetry with respect to the global $\pi$ rotation of spins about the $x$-axis ($S_{\alpha,j}^{y}\mapsto-S_{\alpha,j}^{y}$ and $S_{\alpha,j}^{z}\mapsto-S_{\alpha,j}^{z}$) is spontaneously broken in these phases.

Our previous work [47] has focused on the nature of the phase transitions in the case of $J_{\perp},J_{4}>0$ and easy-axis anisotropy $\Delta>1$, which corresponds to the upper half of Fig. 2. The Néel-SD transition is especially intriguing as it is between two ordered phases breaking different symmetries and may be viewed as a one-dimensional (1D) variant of deconfined quantum critical point [49, 50, 51, 52]. We have presented evidence that this transition belongs to the Gaussian universality class with $c=1$. We have also demonstrated that the RS-Néel transition belongs to the Ising universality class with $c=1/2$.

Our extended analysis in the present work demonstrates that the Gaussian and Ising transition lines found in Ref. [47] cross in the isotropic case $\Delta=1$ and further continue to the easy-plane regime $\Delta<1$, as shown in Fig. 2. We also analyze the case of $J_{\perp},J_{4}<0$, and find similar crossing of the Gaussian and Ising transition lines, as shown in Fig. 3. In both the cases, the Gaussian transitions occur between featureless phases in the easy-plane regime $\Delta<1$, giving examples of topological phase transitions.

It is interesting to ask in what way the four featureless phases are distinguished. The following two types of string order parameters have been used to characterize the Haldane and RS phases [29, 30, 31, 32]:

	$\displaystyle O_{\mathrm{odd}}^{z}(r):=-\bigg{\langle}$	$\displaystyle\quantity(S_{1,0}^{z}+S_{2,0}^{z})\exp\quantity[i\pi\sum_{j=1}^{r-1}\quantity(S_{1,j}^{z}+S_{2,j}^{z})]$
		$\displaystyle\times\quantity(S_{1,r}^{z}+S_{2,r}^{z})\bigg{\rangle},$		(5a)
	$\displaystyle O_{\mathrm{even}}^{z}(r):=-\bigg{\langle}$	$\displaystyle\quantity(S_{1,0}^{z}+S_{2,1}^{z})\exp\quantity[i\pi\sum_{j=1}^{r-1}\quantity(S_{1,j}^{z}+S_{2,j+1}^{z})]$
		$\displaystyle\times\quantity(S_{1,r}^{z}+S_{2,r+1}^{z})\bigg{\rangle}.$		(5b)

Physically, these string order parameters detect the parity of the number $Q$ of valence bonds cut by a vertical line between two adjacent rungs [32]. We obtain $\lim_{r\rightarrow\infty}O_{\text{odd/even}}^{z}(r)\neq 0$ if $Q$ is odd/even. Therefore, the Haldane and RS phases are characterized by non-vanishing of $O_{\mathrm{odd}}^{z}(r)$ and $O_{\mathrm{even}}^{z}(r)$ for $r\to\infty$, respectively. As these order parameters remain unchanged under the operation of $U_{1}^{z}(\pi)$, the Haldane* and RS* phases are characterized similarly. However, these string order parameters do not distinguish between the Haldane and Haldane* phases or between the RS and RS* phases. We therefore calculate topological indices [10, 20, 21, 22] associated with $D_{2}\times\sigma$ and the translational symmetry, and demonstrate that these indices can distinguish all the four featureless phases.

The rest of this paper is organized as follows. In Sec. II, we present a field-theoretical analysis for weak inter-chain couplings, and derive the phase diagrams in Figs.  2 and 3. In Secs. III and IV, we present numerical results for $J_{\perp}=1$ and $J_{\perp}=-1$, respectively. In Sec. V, we analyze topological indices that distinguish the four featureless phases. In Sec. VI, we present a summary and an outlook for future studies. In Appendix A, we apply the field-theoretical formulation in Sec. II to an XYZ ladder studied by Liu et al. [33], and provide a qualitative description of their numerical results. In Appendices B and C, we provide some supplemental numerical results.

For weak inter-chain couplings with $|J_{\perp}|,|J_{4}|\ll 1$, the ground-state phase diagram of the model (I) can be studied by means of effective field theory based on bosonization [53, 54]. In our recent work [47], we have applied this formalism to study the case of $J_{\perp}>0$ and $\Delta\geq 1$, i.e., the upper half of Fig. 2. Here we present a more detailed analysis that reveals the rich phase diagrams in Figs. 2 and 3. Our formulation is an extension of those in Refs. [26, 27, 28, 39, 44, 45, 46], and we take similar notations as those in Refs. [44, 47].

We have performed numerical calculations directly for the infinite system by applying the VUMPS algorithm [37, 38]. In this algorithm, a variational state is prepared in the form of a uniform matrix product state (MPS), and the ground state is obtained by iteratively optimizing constituent tensors to lower the variational energy. In applying it to the present ladder system, we regard two sites on each rung as a single effective site with the local Hilbert space dimension of four. In this section and the next, we adopt the two-site unit cell implementation in Ref. [37] as all the phases discussed in Sec. II have the unit cell consisting of at most two effective sites (i.e., two rungs). To ensure the sufficient convergence, we only used the data points that have the gradient norm $\|B\|<10^{-10}$, where $B$ is the gradient of energy per site with respect to the elementary tensor in VUMPS. The same method was employed in our previous work [47].

In this section, we fix $J=J_{\perp}=1$ and study the ground-state phase diagram in the $J_{4}$-$\Delta$ plane. The obtained phase diagram is shown in Fig. 4, which is qualitatively consistent with the field-theoretical prediction in Fig. 2. As the case of $\Delta\geq 1$ has been analyzed previously [42, 47], we focus on the easy-plane regime $\Delta<1$ throughout this section. Below we explain how the RS-Haldane*-SD phase boundaries are obtained and how the critical properties at these transitions are characterized in our numerical analysis.

In this section, we fix $J=1$ and $J_{\perp}=-1$. The obtained phase diagram in Fig. 10 is qualitatively consistent with the field-theoretical prediction in Fig. 3. Below we explain how the phase boundaries are obtained and how the critical properties are characterized in our numerical analysis.

In the preceding sections, we have used two types of string correlations (5) in analyzing the RS-Haldane* and RS*-Haldane transitions. However, these correlations cannot be used to distinguish between the RS and RS* phases or between the Haldane and Haldane* phases. In this section, we demonstrate that topological indices [10, 20, 21, 22] associated with the $D_{2}\times\sigma$ symmetry and the translational symmetry can distinguish all the four featureless phases. Below we briefly review the basic formalism for classifying 1D SPT phases and calculating topological indices (especially, those of Pollmann and Turner [20]), and then apply it to the present ladder system. While the full classification of SPT phases in a spin-$\frac{1}{2}$ ladder with the $D_{2}\times\sigma$ symmetry has been achieved by Liu et al. [33], our results illustrate how those phases are identified unambiguously in numerical calculations.

Classification of 1D SPT phases can conveniently be discussed using the MPS representation [10, 11, 21, 22, 23]. Assuming the translational invariance, we represent the (normalized) ground state of the infinite system in the form of a canonical MPS as

$\displaystyle\ket{\Psi}=\sum_{\dots,l,m,n,\dots}[\dots\Lambda\Gamma_{l}\Lambda\Gamma_{m}\Lambda\Gamma_{n}\dots]\ket{\dots l,m,n,\dots}.$

(16)

Here, $\Gamma_{m}$ is a $\chi$-by-$\chi$ matrix with $m$ being the spin state at a site, and $\Lambda=\mathrm{diag}(\lambda_{1},\dots,\lambda_{\chi})$ is a diagonal matrix comprised of the Schmidt values associated with a bipartition of the system into half-infinite chains. In our application to the spin-$\frac{1}{2}$ ladder, $m$ runs over the four spin states on a rung. Suppose that $\ket{\Psi}$ is invariant under an on-site unitary transformation, which is represented in the spin basis as a unitary matrix $\Sigma_{mm^{\prime}}$ acting on every site. Namely, we have $|\langle\Psi|\tilde{\Psi}\rangle|=1$, where $|\tilde{\Psi}\rangle$ is the state after the transformation. Then the $\Gamma_{m}$ matrices can be shown to satisfy [65]

$\displaystyle\sum_{m^{\prime}}\Sigma_{mm^{\prime}}\Gamma_{m^{\prime}}=e^{i\theta_{\Sigma}}U_{\Sigma}^{\dagger}\Gamma_{m}U_{\Sigma},$

(17)

where $e^{i\theta_{\Sigma}}$ is a phase factor and $U_{\Sigma}$ is a $\chi$-by-$\chi$ unitary matrix which commutes with $\Lambda$. The phase factors $\{e^{i\theta_{\Sigma}}\}$ form a 1D representation of the symmetry group. The matrices $\{U_{\Sigma}\}$ form a $\chi$-dimensional projective representation of the symmetry group. Namely, it may differ from the linear representation by phase factors:

$$U_{\Sigma_{1}}U_{\Sigma_{2}}=e^{i\rho(\Sigma_{1},\Sigma_{2})}U_{\Sigma_{1}\Sigma_{2}}.$$

(18)

The phases $\rho(\Sigma_{1},\Sigma_{2})$, called the factor set of the representation, can be used to classify different phases. Specifically, if these phases cannot be gauged away by redefining the phases of $U_{\Sigma}$, the state $\{\ket{\Psi}\}$ belongs to a nontrivial SPT phase. When the translational symmetry is further imposed, $e^{i\theta_{\Sigma}}$ can also be used to classify phases [22].

To obtain the matrix $U_{\Sigma}$ and the phase factor $e^{i\theta_{\Sigma}}$ numerically, we introduce a generalized transfer matrix

$\displaystyle T^{\Sigma}_{\alpha\alpha^{\prime};\beta\beta^{\prime}}:=\sum_{m}\quantity(\sum_{m^{\prime}}\Sigma_{mm^{\prime}}\Gamma_{m^{\prime},\alpha\beta})\quantity(\Gamma_{m,\alpha^{\prime}\beta^{\prime}})^{*}\lambda_{\beta}\lambda_{\beta^{\prime}}.$

(19)

Let $X$ be the right eigenvector with the largest absolute eigenvalue $\epsilon^{\Sigma}_{1}$:

$\displaystyle T^{\Sigma}_{\alpha\beta;\alpha^{\prime}\beta^{\prime}}X_{\beta\beta^{\prime}}=\epsilon^{\Sigma}_{1}X_{\beta\beta^{\prime}},$

(20)

where $X$ is normalized such that $XX^{\dagger}=I$. Since we assumed $|\langle\Psi|\tilde{\Psi}\rangle|=1$, $\epsilon^{\Sigma}_{1}$ must have unit modulus. Conversely, if $|\epsilon^{\Sigma}_{1}|<1$, we have $|\langle\Psi|\tilde{\Psi}\rangle|=0$. We now regard the vector $X$ with $\chi^{2}$ elements as a $\chi$-by-$\chi$ matrix. One can show that if $|\epsilon^{\Sigma}_{1}|=1$, $X$ and $\epsilon^{\Sigma}_{1}$ are equal to $U^{\dagger}$ and $e^{i\theta_{\Sigma}}$, respectively, specifically, $U_{\beta\beta^{\prime}}=X_{\beta^{\prime}\beta}^{*}$ [20, 65].

We now focus on the case of a spin-$\frac{1}{2}$ ladder with the $D_{2}\times\sigma$ symmetry studied by Liu et al. [33]. The symmetry transformations acting on each rung are given by the spin rotations $\Sigma^{x}:=\exp[i\pi(S^{x}_{1}+S^{x}_{2})]$ and $\Sigma^{z}:=\exp[i\pi(S^{z}_{1}+S^{z}_{2})]$ and the inter-chain exchange $\Sigma^{\sigma}:=\sum_{\alpha,\beta=\uparrow,\downarrow}\ket{\alpha\beta}\bra{\beta\alpha}$. For these transformations, we can introduce the unitary matrices $U_{x}:=U_{\Sigma^{x}}$, $U_{z}:=U_{\Sigma^{z}}$, and $U_{\sigma}:=U_{\Sigma^{\sigma}}$ and the corresponding indices $\epsilon_{1}^{x}$, $\epsilon_{1}^{z}$, and $\epsilon_{1}^{\sigma}$ as explained above. While $\Sigma^{x}$, $\Sigma^{z}$, and $\Sigma^{\sigma}$ commute with one another, the commutation relations among $U_{x}$, $U_{z}$, and $U_{\sigma}$ may involve nontrivial phase factors. Such phase factors can be used as fingerprints of different phases. They can conveniently be detected by introducing traced commutators [20]

$$\mathcal{O}_{xz}:=\begin{cases}0&\text{if}~{}~{}|\epsilon_{1}^{x}|<1~{}\text{or}~{}|\epsilon_{1}^{z}|<1;\\ \frac{1}{\chi}\tr(U_{x}U_{z}U_{x}^{\dagger}U_{z}^{\dagger})&\text{if}~{}~{}|\epsilon_{1}^{x}|=|\epsilon_{1}^{z}|=1,\end{cases}$$

(21)

and

$$\mathcal{O}_{x\sigma}:=\begin{cases}0&\text{if}~{}~{}|\epsilon^{x}|<1~{}\text{or}~{}|\epsilon_{1}^{\sigma}|<1;\\ \frac{1}{\chi}\tr(U_{x}U_{\sigma}U_{x}^{\dagger}U_{\sigma}^{\dagger})&\text{if}~{}~{}|\epsilon_{1}^{x}|=|\epsilon_{1}^{\sigma}|=1.\end{cases}$$

(22)

As explained above, we have $|\epsilon_{1}^{\Sigma}|<1$ if the state is not invariant under the transformation $\Sigma$; in this case, $\mathcal{O}_{\Sigma\Sigma^{\prime}}$ is zero by definition.

The commutation relations among $U_{x}$, $U_{z}$, and $U_{\sigma}$ can be read off from Table I in Ref. [33], in which possible projective representations of $D_{2}\times\sigma$ and the corresponding SPT phases are summarized. If we do not impose the translational symmetry, the RS and RS* phases (termed rung-$\ket{0,0}$ and rung-$\ket{1,z}$ in Ref. [33]) fall into the trivial phase as they both include rung-factorized product states. In this trivial phase, $U_{x}$, $U_{z}$, and $U_{\sigma}$ commute with one another, indicating $\mathcal{O}_{xz}=\mathcal{O}_{x\sigma}=1$. In contrast, the Haldane and Haldane* phases (termed $t_{0}$ and $t_{z}$ in Ref. [33]) show the nontrivial relation $U_{x}U_{z}=-U_{z}U_{x}$, indicating $\mathcal{O}_{xz}=-1$. Furthermore, the Haldane* phase shows $U_{x}U_{\sigma}=-U_{\sigma}U_{x}$, indicating $\mathcal{O}_{x\sigma}=-1$. If we further impose the translational symmetry, the RS and RS* phases can be distinguished by the index $\epsilon^{\sigma}_{1}$. Indeed, by multiplying $\Sigma_{\sigma}$ to a singlet $\ket{0,0}:=\left(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow}\right)/\sqrt{2}$ and a twisted singlet $\ket{1,z}:=\left(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow}\right)/\sqrt{2}$ on a rung, we find $\epsilon^{\sigma}_{1}=-1$ and $\epsilon^{\sigma}_{1}=1$ in the RS and RS* phases, respectively. Therefore, we can distinguish the four featureless phases by using the indices $\mathcal{O}_{xz}$, $\mathcal{O}_{x\sigma}$, and $\epsilon^{\sigma}_{1}$.