SI.pdf

# Twice Hidden String Order and Competing Phases in the Spin-1/2 Kitaev-Gamma
Ladder

Erik S. Sørensen  0000-0002-5956-1190 Department of Physics and Astronomy,
McMaster University, Hamilton, Ontario L8S 4M1, Canada  Hae-Young Kee∗
0000-0002-8248-4461 Department of Physics, University of Toronto, Ontario M5S
1A7, Canada  Canadian Institute for Advanced Research, CIFAR Program in
Quantum Materials, Toronto, ON M5G 1M1, Canada

###### Abstract

Finding the Kitaev spin liquid in candidate materials involves understanding
the entire phase diagram, including other allowed interactions. One of these
interactions, called the Gamma ($\Gamma$) interaction, causes magnetic
frustration and its interplay with the Kitaev ($K$) interaction is crucial to
comprehend Kitaev materials. Due to the complexity of the combined K$\Gamma$
model, quasi-one dimensional models have been investigated. While several
disordered phases are found in the 2-leg ladder, the nature of the phases are
yet to be determined. Here we focus on the disordered phase near the
antiferromagnetic $\Gamma$ limit (denoted by A$\Gamma$ phase) next to the
ferromagnetic Kitaev phase. We report a distinct non-local string order
parameter characterizing the A$\Gamma$ phase, different from the string order
parameter in the Kitaev phase. This string order parameter becomes evident
only after two unitary transformation, referred to as a twice hidden string
order parameter. The related entanglement spectrum, edge states, magnetic
field responses, and the symmetry protecting the phase are presented, and its
relevance to the two-dimensional Kitaev materials is discussed. Two newly
identified disordered phases in the phase diagram of K$\Gamma$ ladder is also
reported.

## Introduction

Over the course of investigating spin $S$=1/2 two-dimensional (2D) honeycomb
Kitaev materials [1, 2, 3, 4, 5], candidates of long-sought quantum spin
liquids, an additional bond-dependent spin exchange term named Gamma
($\Gamma$) interaction [6] was found along with the bond-dependent Kitaev
($K$) interaction [7, 8]. Unlike the standard Heisenberg interaction, both the
Kitaev and the $\Gamma$ interactions are highly nontrivial and extremely
frustrated. While the same sign of $K$ and $\Gamma$ cancels the frustration
leading to magnetically ordered phases, the regions with different signs of
these interactions are highly frustrated.[5] Since these two interactions are
known to dominate over other symmetry-allowed interactions in the emerging
candidate material $\alpha$-RuCl3, the combined Kitaev-Gamma (K$\Gamma$) model
has attracted considerable attention in the theoretical community and it is
widely accepted [9, 10, 11] that a realistic description of $\alpha$-RuCl3
should be sought in the regime with antiferromagnetic (AFM) Gamma
($\Gamma\mathord{>}$0) and ferromagnetic (FM) Kitaev ($K\mathord{<}$0)
interactions. Understanding the phases that arise in this frustrated regime of
the 2D honeycomb K$\Gamma$-model is therefore of crucial importance.

This particular region of the K$\Gamma$-model has been extensively studied by
a range of numerical methods [5]. Despite detailed studies, the nature of the
phase next to the FM Kitaev spin liquid arising due to AFM $\Gamma$
interaction remains controversial. Most studies have found that it is in a
disordered phase [12, 13, 14, 15, 16, 17, 18, 19], denoted by K$\Gamma$SL for
K$\Gamma$ spin liquid, or nematic paramagnets, but functional renormalization
approaches found magnetically ordered phases [20] while variational Monte
Carlo calculations [21] observed a narrow disordered phase next to the FM
Kitaev spin liquid with most of the antiferromagnetic $\Gamma$ region
dominated by a zig-zag ordered phase.

Motivated by such discrepancy, another approach to investigate the 2D limit of
the K$\Gamma$ model was taken by starting from low-dimensional models with the
hope of furthering the understanding of the honeycomb model by determining the
phases of $n$-leg (brick-wall) models. Despite the obvious challenge in
connecting the two limits, it is reasonable to expect that potential spin
liquid phases arising in the honeycomb model should correspond to regions
where the $n$-leg models display disordered phases. Such an approach was
employed earlier for the pure Kitaev model [22]. Disordered phases in the
anisotropic Kitaev 1-leg chain were found, and they were characterized by non-
local string order parameters (SOPs)[22]. It has also been shown that the
isotropic Kitaev 2-leg ladder model exhibits a disordered phase, characterized
by an unconventional SOP different from that of the anisotropic chain Kitaev
phase [23].

The one-dimensional (1D) chain and ladder version of the K$\Gamma$ model were
investigated numerically with very high precision, as the reduced
dimensionality allows accessing bigger system sizes [24, 25, 26, 27, 28]. In
the 1D chain model, it was found that the pure Gamma model belongs to a
Luttinger liquid phase governed by the gapless hidden SU(2) Heisenberg chain,
a fact revealed after a 6-site transformation, i.e, a duality mapping [29,
24]. A study of the same K$\Gamma$ model on a quasi 1D ladder using DMRG and
iDMRG techniques[18, 30] found a magnetically disordered phase, possessing a
small gap near the AFM pure Gamma limit. This phase surrounding the pure AFM
Gamma point next to the FM Kitaev phase (denoted by F$K$) was referred to as
the A$\Gamma$ phase. Even though the A$\Gamma$ phase in the ladder occurs in
the same part of the phase diagram as the proposed $K\Gamma$SL in the 2D
limit, a distinct name was introduced, as it is yet to be determined how the
A$\Gamma$ phase is connected to the 2D limit $K\Gamma$SL.

While it is clear that there is no magnetic order in this phase, the precise
nature of the A$\Gamma$ phase has not yet been settled due to its complex
nature. The presence of a gap indicates that the A$\Gamma$-phase is likely a
symmetry protected topological (SPT) phase [31, 32, 33]. If so, it is of
importance to identify a corresponding SOP, edge states, and the symmetry that
protects this phase, which are characteristic of the SPT, and to determine how
this phase respond to an external magnetic field. If strong evidence for a
non-trivial SPT nature of the A$\Gamma$ phase can be established, it would
establish a next step to the proposed $K\Gamma$SL in the 2D limit. In the
following sections, we will systematically examine these inquiries and provide
comprehensive responses. To perform a thorough analysis, we start by reviewing
the full phase diagram prior to focusing on the nature of the A$\Gamma$ phase.
As shown below, we also report two other disordered phases.

## Results

### Model

Figure 1: The K$\Gamma$ ladder and corresponding phase diagram. a A strip of
the K$\Gamma$ honeycomb lattice corresponding to a two-leg K$\Gamma$ ladder
with alternating $x$ and $y$ bonds along the leg and $z$-bond between the
chains, with the numbering of the sites used throughout the paper. The dotted
$z$-bonds arise from imposing periodic boundary conditions in the direction
perpendicular to the ladder. However, all $z$-bonds are taken to have equal
strength. The dashed red line indicates the partition used for $\rho_{B}$
while the dashed blue line indicates the partition used for $\rho_{A}$. b
Schematic phase diagram of the K$\Gamma$ ladder, Eq. (1), for
$\Gamma\mathord{>}$0\.

The K$\Gamma$ Hamiltonian is given by

$H_{K\Gamma}=\sum_{\langle
i,j\rangle_{\gamma\in(x,y,z)}}KS_{i}^{\gamma}S_{j}^{\gamma}+\Gamma\left(S^{\alpha}_{i}S^{\beta}_{j}+S^{\beta}_{i}S^{\alpha}_{j}\right)$
(1)

where $(\alpha,\beta)$ takes on the values $(y,z)/(x,z)/(x,y)$ for
$\gamma=x/y/z$, and $\langle i,j\rangle$ refers to the nearest neighbor sites.
An alternative representation of the honeycomb lattice is as a brick-wall
lattice [22], and its two-leg limit with periodic boundary conditions is
simply a ladder shown in Fig. 1a. The dotted bonds indicate Kitaev $z$-bonds
arising from periodic boundary conditions, and we shall always take such bonds
to be identical to the regular (solid) $z$-bonds, in which case the honeycomb
strip can be viewed as a regular rectangular ladder.

We parameterize the model by taking

$K=\cos\phi,\ \ \ \mathrm{and}\ \ \ \ \Gamma\mathord{=}\sin\phi,$ (2)

and interpolate between the Kitaev and $\Gamma$ interactions by varying
$\phi$. Our main interest is in the region with $\phi/\pi\in[0,\pi]$ where
$\Gamma\mathord{>}$0 and the Kitaev term, $K$, changes from AFM to FM at
$\phi$=$\pi$/2, as this region is relevant to most two-dimensional (2D) Kitaev
candidate materials. The total number of sites in the ladder (including both
legs) is denoted by $N$.

### Phase Diagram

A full phase diagram is shown in Fig. 1b. It is obtained by various quantities
presented in the next subsection. Moving from $\phi\mathord{=}0$ to $\pi$, the
AFM Kitaev phase (denoted by AK), a FM phase denoted by FM${}_{U_{6}}$,
A$\Gamma$, and F$K$ phases are found consistent with the earlier works[18,
30]. At the special point $\phi=\pi/4$, the FM${}_{U_{6}}$ phase can be mapped
to the ferromagnetic Heisenberg ladder by applying a local unitary $U_{6}$
transformation (see Supplementary Note 1), and is therefore gapless. However,
for $\phi\mathord{\neq}\pi/4$ a small gap appear, as we show in Supplementary
Note 3. Surprisingly, two additional phases denoted by SPTα and SPTβ can be
identified between the FM${}_{U_{6}}$ and A$\Gamma$ phases. As we will show
below, they are magnetically disordered and display the characteristics of SPT
phases, i.e., doubled entanglement spectrum. Beyond $\phi=\pi$, the rung
singlet phase denoted by RS${}_{U_{6}}$ phase delineates the F$K$ phase.

In addition to the expected Kitaev phases A$K$ and F$K$ the appearance of the
FM${}_{U_{6}}$ and RS${}_{U_{6}}$ phases are well established in the K$\Gamma$
honeycomb and n-leg models. After the local $U_{6}$ transformation [29],
corresponding to local spin rotations, at $\phi\mathord{=}\pi/4$ and $5\pi/4$,
i.e., $K$=$\Gamma$, the K$\Gamma$ 2D honeycomb model is equivalent to the FM
and AFM Heisenberg model, respectively. The application of the $U_{6}$
transformation is specified in Supplementary Note 1. To understand the nature
of the other three phases, A$\Gamma$, SPTα and SPTβ, we first performed a
detailed analysis of the entanglement spectrum.

Figure 2: Entanglement spectrum and susceptibility for
$\phi/\pi\mathord{<}0.5$. iDMRG results for: a Entanglement spectrum from
$\rho_{A}$, red dashed line in Fig. 1a. b Entanglement spectrum from
$\rho_{B}$, blue dashed line in Fig. 1a. The numbers refer to the degeneracy
of the eigenvalue. c $\chi^{e}_{\phi}$. Two distinct phases, SPTα and SPTβ,
are visible between the FM${}_{U_{6}}$ and A$\Gamma$ phase. The red shading
between $\phi/\pi$=0.428-0.442 denotes a transitional region of limited
convergence due to a field instability.  Figure 3: Entanglement spectrum and
susceptibility for $\phi/\pi\mathord{>}0.5$. iDMRG results for: a Entanglement
spectrum from $\rho_{A}$, red dashed line in Fig. 1a. b Entanglement spectrum
from $\rho_{B}$, blue dashed line in Fig. 1a. The numbers refer to the
degeneracy of the eigenvalue. c $\chi^{e}_{\phi}$. The A$\Gamma$ F$K$ and
RS${}_{U_{6}}$ phases are clearly delineated.

### Entanglement Spectrum

Our results for the entanglement spectrum, as well as for the susceptibility,
$\chi^{e}_{\phi}$ are shown in Figs. 2 and 3. Due to the complexity of the
phase diagram, we split it into two regions centered around the A$\Gamma$
phase: Fig. 2 represents the left part of the A$\Gamma$ and Fig. 3 the right
part of the A$\Gamma$ phase. We consider two different partitions shown in
Fig. 1a corresponding to $\rho_{A}$ and $\rho_{B}$. In both Fig. 2 and Fig. 3
iDMRG results are shown for both partitions with $\rho_{A}$ shown in panel (a)
and $\rho_{B}$ in panel (b) with $\chi^{e}_{\phi}$ in panel (c).

To the left of the A$\Gamma$-phase, two other phases SPTα and SPTβ are clearly
separated from the FM${}_{U_{6}}$ and A$\Gamma$, as noted from their
entanglement spectrum for $\rho_{A}$. Furthermore, the double degeneracy of
the entanglement spectrum shown in Fig. 2 (b) from $\rho_{B}$ is a clear
signal of a SPT phase [34, 35, 36, 37, 33]. Note that there is only a weak
signature in $\chi^{e}_{\phi}$ of the transition between the two phases,
visible at $\phi$=$0.3956\pi$ in Fig. 2(c), corresponding to a small
discontinuity in $\chi^{e}_{\phi}$. While the FM${}_{U_{6}}$ phase appears
abruptly at $\phi$=$0.3823\pi$, the transitions between the A$\Gamma$ phase
and SPTβ phase at $\phi\sim 0.435\pi$ is not discernible in $\chi^{e}_{\phi}$.
The blurriness of the transition is likely due to a field induced phase that
pinches off to a single point at zero field at the A$\Gamma$-SPTβ transition,
thereby obscuring it. We note that, the quantum critical points (QCPs) are
immediately noticeable in the entanglement spectra.

Moving to the right within the A$\Gamma$ phase which encompass the point
$\Gamma$ =1, the transition to the F$K$ phase from the A$\Gamma$ phase occurs
at $\phi\mathord{=}0.88271(5)\pi$, which is clearly visible in the
entanglement spectra as well as $\chi^{e}_{\phi}$ as shown in in Fig. 3. The
transition between the RS${}_{U_{6}}$ and FK phases at $\phi$=$\pi$ is less
apparent in the structure of the entanglement spectrum in Fig. 3 (a) and (b).
However, from the results for $\chi^{e}_{\phi}$ in Fig. 3(c) this transition
is immediately visible, and it was also noted that the second derivative of
$\lambda_{1}$ clearly detects the transition [30].

Similar to the A$K$ phase, none of the phases SPTα, SPTβ, A$\Gamma$ and F$K$
has any long-range magnetic ordering, nor is there any indication of nematic
(quadropolar) or chiral ordering. As discussed in Supplementary Note 2, all
four phases are gapped with a finite sizeable correlation length. The
difference between them is captured in the entanglement spectrum. In the SPTα,
SPTβ and F$K$ (and A$K$) phases, the entanglement spectrum have all entries
doubled when considering $\rho_{B}$. For the A$\Gamma$-phase, the same applies
to the spectrum for $\rho_{A}$. Since an entanglement spectrum where all
eigenvalues have degeneracy larger than one is a signature of a topological
non-trivial phase, in the next section, we investigate the projective symmetry
analysis to confirm their non-trivial topology prior to presenting associated
non-local SOPs.

### Projective Symmetry Analysis

With the SPTα, SPTβ, A$\Gamma$ and F$K$ phases as potential SPT phases, it is
of considerable interest to investigate the projective representations [37,
38, 39, 40, 41, 42, 33] $U$, of a site symmetry $\mathcal{R}$. This has been
done for $S$=1 chains [37, 38, 39, 40, 41, 42] and ladders [43] and also for
$S$=1/2 ladders [41, 44, 45, 46, 47]. The matrices $U$ can be obtained from
the generalized transfer matrices in iDMRG calculations as described in Ref.
48.

For the K$\Gamma$ ladder, it is a significant simplification to consider the
transformed model obtained after applying the $U_{6}$ transformation. This
transformation maps the original Hamiltonian $H_{\mathrm{K}\Gamma}$ to the
transformed ladder, denoted by $H_{\mathrm{K}\Gamma}^{U_{6}}$. Under the
$U_{6}$ transformation the $x$, $y$ and $z$-bonds of the K$\Gamma$-ladder are
transformed into anisotropic Heisenberg bonds, $x^{\prime}$, $y^{\prime}$ and
$z^{\prime}$ in the following manner:

$\displaystyle x^{\prime}\ :\
-KS^{x}_{i}S^{x}_{j}-\Gamma(S^{y}_{i}S^{y}_{j}+S^{z}_{i}S^{z}_{j})$
$\displaystyle y^{\prime}\ :\
-KS^{y}_{i}S^{y}_{j}-\Gamma(S^{x}_{i}S^{x}_{j}+S^{z}_{i}S^{z}_{j})$
$\displaystyle z^{\prime}\ :\
-KS^{z}_{i}S^{z}_{j}-\Gamma(S^{x}_{i}S^{x}_{j}+S^{y}_{i}S^{y}_{j})$ (3)

In the transformed ladder $H_{\mathrm{K}\Gamma}^{U_{6}}$ model, the definition
of $x^{\prime}$, $y^{\prime}$ and $z^{\prime}$-bonds, can pictorially be
represented as shown in Fig. 4(a).

Figure 4: Unit cells of the $U_{6}$ transformed ladder. Two unit cells of
$H_{\mathrm{K}\Gamma}^{U_{6}}$, the K$\Gamma$ ladder after the $U_{6}$ local
transformation with $N\mathord{=}6n$ sites. a Two regular unit cells with
regular open boundary conditions (OBC). b Two slanted unit cells with slanted
OBC.

See Supplementary Note 1. We will consider two different open boundary
conditions (OBC) as shown in Fig. 4. The unit cells depicted in panel (a) are
referred to as regular unit cells with regular OBC, while the ones in panel
(b) are referred to as slanted unit cells with slanted OBC.

In order to understand the symmetries of the matrix product state (MPS) wave-
function, it is useful to write it in the canonical form [49, 50, 51, 52]:

$|\Psi\rangle=\sum_{j_{1},\ldots,j_{N}}M^{[1]}_{j_{1}}\Lambda^{[2]}M^{[2]}_{j_{2}}\ldots\Lambda^{[N]}M^{[N]}_{j_{N}}|j_{1},\ldots,j_{N}\rangle,$
(4)

where the $M^{[n]}_{j_{n}}$ are complex matrices and the $M^{[n]}$, real,
positive, square diagonal matrices. In the iDMRG formulation, the set of
matrices on any unit cell becomes the same $M^{[n]}_{j}$=$M_{j}$, $M^{[n]}$ =
$M$ for all $n$, although they may vary within the unit cell. For the
translationally invariant state, it can be shown [53, 34] that for any (site)
symmetry operation $g$, represented in the spin basis by the unitary matrix,
$\Sigma_{jj^{\prime}}(g)$, the $M_{j}$ matrices must transform as [34, 48]:

$\sum_{j^{\prime}}\Sigma_{jj^{\prime}}(g)M_{j^{\prime}}=e^{i\theta}U^{\dagger}(g)M_{j}U(g),$
(5)

where the unitary matrix $U(g)$ commutes with the $M$ matrices and
$e^{i\theta}$ is a phase factor. With $D$ denoting the bond dimension, the $U$
matrices form a $D$-dimensional projective representation of the symmetry
group of the wave-function, and they can be determined from the unique
eigenvector of the generalized transfer matrix [34, 48] with eigenvalue
$|\lambda|$=1, where the generalized transfer matrix is defined as

$T^{\Sigma}_{\alpha\alpha^{\prime};\beta\beta^{\prime}}=\sum_{j}\left(\sum_{j^{\prime}}\Sigma_{jj^{\prime}}M_{j^{\prime},\alpha\beta}\right)(M_{j,\alpha^{\prime}\beta^{\prime}})^{*}\Lambda_{\beta}\Lambda_{\beta^{\prime}}$
(6)

If the largest eigenvalue is $|\lambda|<$1, the symmetry is not a property of
the state being considered. The projective representation is reflected in the
fact that if $\Sigma(g)\Sigma(h)$=$\Sigma(gh)$, then

$U(g)U(h)=e^{i\phi(g,h)}U(gh),$ (7)

where the phases $\phi(g,h)$ are characteristic of the topological phase.

Let us now consider the site symmetries, $\mathcal{R}^{x}$ and
$\mathcal{R}^{y}$ defined as:

$\mathcal{R}^{x}_{l}=e^{i\pi S^{x}_{l}}\ ,\ \ \ \mathcal{R}^{y}_{l}=e^{i\pi
S^{y}_{l}}\ ,\ \ \ \mathcal{R}^{z}_{l}=e^{i\pi S^{z}_{l}}.$ (8)

The Hamiltonian $H_{\mathrm{K}\Gamma}^{U_{6}}$ is invariant under the
operators $\prod_{l}\mathcal{R}^{\gamma}_{l}$, with $\gamma\mathord{=}x,y,z$
with distinct quantum numbers for the low-lying states. If these symmetries
are respected, their representations can differ by a phase,
$\phi(\mathcal{R}_{x},\mathcal{R}_{y})$ that must obey
$e^{i\phi(\mathcal{R}_{x},\mathcal{R}_{y})}$=$\pm 1$:

$U(\mathcal{R}^{x})U(\mathcal{R}^{y})=\pm
U(\mathcal{R}^{y})U(\mathcal{R}^{x}).$ (9)

Furthermore, the non-trivial value
$e^{i\phi(\mathcal{R}_{x},\mathcal{R}_{y})}$=-1 can only occur if all
eigenvalues of the entanglement spectrum are at least twice degenerate [34].
The phase factor can then be isolated by defining [48]:

$\mathcal{O}_{\mathrm{Z_{2}\times
Z_{2}}}\equiv\frac{1}{D}\operatorname{Tr}\left(U(\mathcal{R}^{x})U(\mathcal{R}^{y})U^{\dagger}(\mathcal{R}^{x})U^{\dagger}(\mathcal{R}^{y})\right),$
(10)

with $D$ the bond dimension, with similar definitions for other pairs of
operators in Eq. (8). In the above definitions it is understood that the
transformations are applied throughout the lattice and in order to obtain the
matrices $U$, generalized transfer matrices representing the relevant unit
cell has to be considered.

The site symmetries, $\mathcal{R}^{x}$, $\mathcal{R}^{y}$ and
$\mathcal{R}^{z}$, forming the dihedral group, $D_{2}$, are respected by both
of the unit cells in Fig. 4(a) and 4(b). Using generalized transfer matrices
obtained from unit cells of the shape shown in Fig 4(a) when studying the
A$\Gamma$-phase and of the slanted shape shown in Fig. 4(b) when studying the
F$K$, SPTα, and SPTβ phases, we obtain

$\mathcal{O}_{\mathrm{Z_{2}\times Z_{2}}}=-1,$ (11)

for the SPTα, SPTβ, A$\Gamma$, and F$K$ phases.

Similar analysis can be made for the time-reversal $(\mathrm{TR})$ symmetry,
defined by $M_{j}\to\sum_{j^{\prime}}\left[e^{i\pi
S^{y}}\right]_{jj^{\prime}}M^{\star}_{j^{\prime}}$, with $\star$ denoting
complex conjugation. In this case, it can be established that [34]
$U_{\mathrm{TR}}U_{\mathrm{TR}}^{\star}$=$e^{i\phi(\mathrm{TR},\mathrm{TR})}\mathbb{1}$
where the phase $\phi(\mathrm{TR},\mathrm{TR})$ cannot be absorbed into the
definition of $U_{\mathrm{TR}}$. One should note that for most other
symmetries, with the notable exception of inversion, similar considerations
will lead to $U^{2}$=$e^{i\phi}\mathbb{1}$ in which case the phase $\phi$ in
fact can be absorbed into the definition of $U$. For instance, this is the
case for $U\left(\mathcal{R}^{\alpha}\right)$ discussed above. However, for
time reversal, the phase factor $e^{i\phi(\mathrm{TR},\mathrm{TR})}$ can
directly be extracted by defining [48]:

$\mathcal{O}_{\mathrm{TR}}\equiv\frac{1}{D}\operatorname{Tr}\left(U_{\mathrm{TR}}U_{\mathrm{TR}}^{\star}\right),$
(12)

and again one finds that $\phi(\mathrm{TR},\mathrm{TR})$=0,$\pi$, so that
$\mathcal{O}_{\mathrm{TR}}$=$\pm$1\. As an example, for the $S$=1 Heisenberg
spin chain in the Haldane phase it is known that
$\mathcal{O}_{\mathrm{Z_{2}\times Z_{2}}}$=$-1$,
$\mathcal{O}_{\mathrm{TR}}$=$-1$ [34, 35].

Using generalized transfer matrices obtained from unit cells of the shape
shown in Fig 4(a) for the A$\Gamma$-phase and of the slanted shape for the
F$K$, SPTα, and SPTβ phases, we obtain

$\mathcal{O}_{\mathrm{TR}}=-1,$ (13)

consistent with the presence of a doubled entanglement spectrum in all cases.
As was the case for $\mathcal{O}_{\mathrm{Z_{2}\times Z_{2}}}$ , if the unit
cells are interchanged, one finds instead $\mathcal{O}_{\mathrm{TR}}$=1.

Table 1: Summary of projective analysis. The superscript $\mathrm{regular}$ refers to the unit cell from Fig. 4a, while the superscript $\mathrm{slanted}$ refers to the unit cell from Fig. 4b. Negative values indicate that the state transforms non-trivially. Phase | $\mathcal{O}_{\mathrm{TR}}^{\mathrm{regular}}$ | $\mathcal{O}_{\mathrm{Z_{2}\times Z_{2}}}^{\mathrm{regular}}$ | $\mathcal{O}_{\mathrm{TR}}^{\mathrm{slanted}}$ | $\mathcal{O}_{\mathrm{Z_{2}\times Z_{2}}}^{\mathrm{slanted}}$
---|---|---|---|---
SPTα | 1 | 1 | -1 | -1
SPTβ | 1 | 1 | -1 | -1
A$\Gamma$ | -1 | -1 | 1 | 1
F$K$ | 1 | 1 | -1 | -1

A summary of our results from the projective analysis are provided in Table 1,
negative values indicate that the state transforms non-trivially. For all 4
phases, it is seen that a unit cell can be chosen for which the state
transforms non-trivially under both the $\mathrm{TR}$ and
$\mathcal{O}_{\mathrm{Z_{2}\times Z_{2}}}$ symmetries.

Based on the above analysis of entanglement spectrum and projective symmetry,
we conclude that A$\Gamma$ phase is an SPT phase. It is then important to
further identify its SOP that differentiates this phase from the other
disordered phases.

### Twice Hidden String Order

To establish a non-local string order parameter (SOP) characterizing the
A$\Gamma$ phase, we need to exploit a non-local unitary transformation that
maps the original Hamiltonian with OBC to a new Hamiltonian that exhibits a
local long-range order [54, 55, 56]. We found that it is difficult to identify
such non-local transformation starting from the original
$H_{\mathrm{K}\Gamma}$, but it can be achieved by first applying the $U_{6}$
transformation to arrive at $H_{\mathrm{K}\Gamma}^{U_{6}}$. It is then
possible to define a non-local unitary transformation $W$, mapping
$H_{\mathrm{K}\Gamma}^{U_{6}}$ to a new local Hamiltonian. We denote the
resulting Hamiltonian, where four-spin terms appear, by $H_{\mathrm{KQ}}$. For
the parameters relevant for the A$\Gamma$-phase, $H_{\mathrm{KQ}}$ exhibits
long-range order in the spin-spin correlation functions, corresponding to a
local order parameter. Due to the application of two separate unitary
transformations, one might consider the resulting order to be twice hidden.

The non-local unitary operator $W$ for a $N$-site ladder with OBC that maps
$H_{\mathrm{K}\Gamma}^{U_{6}}$ to $H_{\mathrm{KQ}}$ takes the following form.

$W=\prod_{\begin{subarray}{c}j+1<k\\\ j\ \mathrm{odd},\ k\ \mathrm{odd}\\\
j=1,\ldots N-3\\\ k=3,\ldots N-1\end{subarray}}w(j,k).$ (14)

With the individual $w(j,k)$ given as follows:

$w(j,k)=e^{i\pi(S^{y}_{j}+S^{y}_{j+1})\cdot(S^{z}_{k}+S^{z}_{k+1})},$ (15)

and $W^{\dagger}$=$W$. The OBC are here crucial for the mapping to be exact.
Evidently, all $w(j,k)$ are unitary and therefore also $W$, and
$\left[w(j,k),w(l,m)\right]=0\ \ \forall\ j,k,l,m$. Note that this is a
different labelling of the unitary operator introduced in Ref. [23, 30]. It
can also be shown that other combinations of the spin operators $S^{\alpha}$
appearing in Eq. (15) lead to equivalent unitary operators, for instance,
$e^{i\pi(S^{y}_{j}+S^{y}_{j+1})\cdot(S^{x}_{k}+S^{x}_{k+1})}$ is another valid
choice. However, the specific choice made in Eq. (15) will influence the type
of ordering that is observed in $H_{\mathrm{KQ}}$, as well as the specific
form of $H_{\mathrm{KQ}}$. Schematically, the transformations can be viewed as
shown in Fig. 5. The detailed form of $H_{\mathrm{KQ}}$ after the $W$
transformation on $H_{\mathrm{K}\Gamma}^{U_{6}}$ is presented in Supplementary
Note 4.

Figure 5: Pictorial view of KG transformations. The two transformations from
the original K$\Gamma$ model to $H_{\mathrm{K}\Gamma}^{U_{6}}$, and the
subsequent transformation to $H_{\mathrm{KQ}}$ are sketched. The type of order
parameter for each Hamiltonian is indicated.  Figure 6: Spin correlations in
the $H_{\mathrm{KQ}}$ model at $\phi\mathord{=}0.85\pi$. DMRG results with
$N$=384 for the correlation function $4\langle S^{\prime\prime
y}(0)S^{\prime\prime y}(r)\rangle$ versus distance, $r$ along one leg of the
ladder. $r$=0 corresponds to site 47. Results are for $H_{\mathrm{KQ}}$ with
$\phi\mathord{=}0.85\pi$ and $r$ is measured in lattice spacings along the
leg.

We first discuss the ordering in $H_{\mathrm{KQ}}$ where we denote the spins
by $S^{\prime\prime\alpha}$, where the double prime represents the two
transformations of spin from the original Hamiltonian. It is then convenient
to study correlation functions of the form $4\langle
S^{\prime\prime\alpha}(0)S^{\prime\prime\alpha}(r)\rangle$ along the legs of
the ladder with $r$ measured in lattice spacings along the leg. To avoid
boundary effects, $r$=0 is usually taken to correspond to a site in the bulk
of the chain. In Fig. 6 we show results for $4\langle S^{\prime\prime
y}(0)S^{\prime\prime y}(r)\rangle$ starting from site 47 with
$\phi\mathord{=}0.85\pi$. Long-range order is clearly present. Similar results
can be obtained for the other leg of the ladder as well as for $4\langle
S^{\prime\prime z}(0)S^{\prime\prime z}(r)\rangle$. However, due to the choice
of spin operators in the definition of $w$ in Eq. (15), there is no ordering
in $4\langle S^{\prime\prime x}(0)S^{\prime\prime x}(r)\rangle$.

Using the inverse of the non-local unitary operator $W$ from Eq. (14) the
above results for $4\langle S^{\prime\prime y}(0)S^{\prime\prime y}(r)\rangle$
is reproduced as a non-local string order correlation function in
$H_{\mathrm{K}\Gamma}^{U_{6}}$ where we denote the spin variables by
$S^{\prime\alpha}$=$\sigma^{\prime\alpha}/2$. We find $\langle
S_{1}^{\prime\prime y}S_{r+1}^{\prime\prime y}\rangle$ is given by

$\displaystyle{\cal O}^{y}(r)$ $\displaystyle=$ $\displaystyle 4\langle
S_{1}^{\prime\prime y}S_{r+1}^{\prime\prime
y}\rangle=(-1)^{\left\lfloor{r/2}\right\rfloor}$ (19) $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{rl}\langle\sigma_{2}^{\prime
y}\left(\prod_{k=3}^{r}\sigma_{k}^{\prime y}\right)\sigma_{r+1}^{\prime
y}\rangle&\mbox{ $r$ even}\\\ \\\ \langle\sigma_{2}^{\prime
y}\left(\prod_{k=3}^{r-1}\sigma_{k}^{\prime y}\right)\sigma_{r+1}^{\prime
y}\rangle&\mbox{ $r$ odd}\end{array}\right.$

Note that, to fully reproduce the results for $H_{\mathrm{KQ}}$ shown in Fig.
6 with the string correlation function in (19) for
$H_{\mathrm{K}\Gamma}^{U_{6}}$, a relabelling of the sites needs to be done
that we have skipped for clarity.

Figure 7: String order parameters in the A$\Gamma$ and surrounding phases.
iDMRG results for the string order parameters in the $H_{\mathrm{K}\Gamma}$
and $H_{\mathrm{K}\Gamma}^{U_{6}}$ models, shown alongside DMRG results for
$H_{\mathrm{KQ}}$. Orange circles, $\Xi^{y}$ for $H_{\mathrm{K}\Gamma}$. The
inset shows $\Xi^{y}(r)$ versus $r$ at $\phi\mathord{=}0.85\pi$ for
$H_{\mathrm{K}\Gamma}$. Magenta circles, DMRG results for ${\cal
O}_{\mathrm{even}}^{z}$ for $H_{\mathrm{KQ}}$. Light blue circles, ${\cal
P}^{z}_{\mathrm{even}}$ for $H_{\mathrm{K}\Gamma}^{U_{6}}$. Green circles,
${\cal Z}$ for $H_{\mathrm{K}\Gamma}^{U_{6}}$.

We can now apply the inverse $U_{6}$ transformation (see Supplementary Note 1)
to the above expressions for ${\cal O}^{y}(r)$ to determine the string order
correlation functions that define the A$\Gamma$-phase in the original
$H_{\mathrm{K}\Gamma}$:

$\Xi^{y}(r)=U_{6}^{-1}({\cal O}^{y}(r)),$ (20)

with the $U_{6}^{-1}$ transformation detailed in Supplementary Note 1. The SOP
in $H_{\mathrm{K}\Gamma}$ Hamiltonian is then given by

$\Xi^{y}=\max_{r\to\infty}\Xi^{y}(r).$ (21)

It is interesting to note that for a small $r$, for example $r=7$
$\Xi^{y}(r=7)$ corresponds to the plaquette operator found in the pure Gamma
model in the honeycomb lattice.[57]

In Fig. 7 we show iDMRG results for $\Xi^{y}$ (orange circles). The inset
shows iDMRG results for $\Xi^{y}(r)$ versus $r$ at $\phi\mathord{=}0.85\pi$
which at large $r$ can be compared to the results for $H_{\mathrm{KQ}}$ shown
in Fig. 6. Note that the results in Fig. 6 show the correlations along a leg,
whereas the inset in Fig. 7 show results from both legs without including the
sign and the relabelling of the sites. Due to the different methods used,
there is not an exact equivalence for very small values of $r$.

We emphasize that the appearance of a non-local SOP in the A$\Gamma$ phase of
$H_{\mathrm{K}\Gamma}$ is equivalent to the presence of long-range order in
$H_{\mathrm{KQ}}$. Hence, $\Xi^{y}$ is non-zero throughout the A$\Gamma$-phase
and goes to zero at the critical points delineating this phase. It is absent
in the other disordered phases, SPTα, SPTβ, and F$K$, and thus uniquely
defines the A$\Gamma$ phase.

With the identification of the A$\Gamma$-phase with regular long-range
ordering in the $H_{\mathrm{KQ}}$ model, it is natural to ask if a regular
(local) order parameter can also be identified for the SPTα\- and SPTβ-phases
in the $H_{\mathrm{KQ}}$. However, all local order parameters that we have
investigated have not shown any ordering in the SPTα\- and SPTβ-phases for
$H_{\mathrm{KQ}}$. Extending the string-order correlation function defined for
the $S\mathord{=}1$ Haldane chain, a heuristic string-order correlation
function has been proposed for $S$=1/2 ladders by pairing two
$S\mathord{=}1/2$ [58, 59]. Following the numbering of Fig. 1a, if
$\tau_{i}^{\alpha}\mathord{=}S^{\alpha}_{2i}+S^{\alpha}_{2i+1}$ are the sum of
two diagonally situated spins, one defines [58, 59]:

${\cal
O}_{\mathrm{even}}^{\alpha}(r)=\langle\tau_{i}^{\alpha}\exp(i\pi\sum_{l=i+1}^{i+r-1}\tau_{l}^{\alpha})\tau_{i+r}^{\alpha}\rangle.$
(22)

The associated SOP is non-zero in the phase surrounding $\phi\mathord{=}0$ in
$H_{\mathrm{KQ}}$ [30], corresponding to the AFM Kitaev (A$K$) phase in
$H_{\mathrm{K}\Gamma}$. The magenta points in Fig. 7 show our results for
${\cal O}_{\mathrm{even}}^{z}$ for the $H_{\mathrm{KQ}}$ model, which clearly
is non-zero in the SPTα\- and SPTβ-phases. This is consistent with the
nonexistence of a local order parameter in these two phases for
$H_{\mathrm{KQ}}$. We note that, due to the heuristic nature of ${\cal
O}_{\mathrm{even}}^{z}$, it is not clear how to associate it with a local
order in a related model. Since $H_{\mathrm{K}\Gamma}$ and
$H_{\mathrm{K}\Gamma}^{U_{6}}$ are related by a local unitary transformation,
any ordering in the SPTα\- and SPTβ-phases in either model would immediately
be apparent in both, and we have not observed any for either model.

Building on the above results for ${\cal O}_{\mathrm{even}}^{z}$ for
$H_{\mathrm{KQ}}$ we propose a closely related heuristic string order
correlation function for $H_{\mathrm{K}\Gamma}^{U_{6}}$ in the following way:
Define $l_{i}^{\alpha}\mathord{=}S^{\alpha}_{2i}-S^{\alpha}_{2i+1}$ as the
difference of two diagonally situated spins, following the numbering from Fig.
1a. We then have

${\cal P}_{\mathrm{even}}^{\alpha}(r)=\langle
l_{i}^{\alpha}\exp(i\pi\sum_{l=i+1}^{i+r-1}\tau_{l}^{\alpha})l_{i+r}^{\alpha}\rangle,$
(23)

with $\tau_{i}^{\alpha}\mathord{=}S^{\alpha}_{i,1}+S^{\alpha}_{i+1,2}$ as
above. Results for ${\cal P}_{\mathrm{even}}^{z}$ versus $\phi$ obtained from
iDMRG calculations with $H_{\mathrm{K}\Gamma}^{U_{6}}$ are shown in Fig. 7 as
the light blue points. The F$K$, SPTα and SPTβ-phases are clearly defined by a
non-zero ${\cal P}_{\mathrm{even}}$. By applying $U_{6}^{-1}$ to the
definition of ${\cal P}_{\mathrm{even}}$ it is straightforward to perform
similar calculations using $H_{\mathrm{K}\Gamma}$ by evaluating
$U_{6}^{-1}({\cal P}_{\mathrm{even}}^{\alpha}(r))$. Even though the definition
${\cal P}_{\mathrm{even}}$ is heuristic, we interpret this result as a
verification of the SPT nature of the F$K$, SPTα\- and SPTβ-phases.

For the F$K$-phase, it is also instructive to consider an even simpler
heuristic string order correlation function, $\mathcal{Z}$ defined as follows
[22]:

$\mathcal{Z}=\langle\prod_{i}^{i+r}\sigma_{i}^{z}\rangle$ (24)

We consider this string correlator for $H_{\mathrm{K}\Gamma}^{U_{6}}$ or
equivalently to $H_{\mathrm{K}\Gamma}$ through application of $U_{6}^{-1}$.
Results for $\mathcal{Z}$ are shown in Fig. 7 versus $\phi$ (green points) as
obtained from iDMRG calculations. Throughout the F$K$-phase $\mathcal{Z}$ is
almost identical to 1, dropping to zero at the transition to the
A$\Gamma$-phase. However, we note that $\mathcal{Z}$ remain sizable throughout
much of the RS${}_{U_{6}}$-phase, reflecting its heuristic nature.

### Edge states and response to magnetic field

Another signature of SPT phases is the presence of edge states under OBC
related to a ground state degeneracy. For the SPT phases in the K$\Gamma$
ladder, it is clear from the degeneracy of the entanglement spectrum that we
need to consider different shapes of clusters (regular vs. slanted OBCs) for
the different SPT phases. In this section, we exclusively study the original
Hamiltonian $H_{\mathrm{K}\Gamma}$ and do not consider
$H_{\mathrm{K}\Gamma}^{U_{6}}$ nor $H_{\mathrm{KQ}}$. For A$\Gamma$ phase, we
use $N\mathord{=}4n$ with the regular OBC and for the remaining SPT phases, we
use the slanted OBC with $N\mathord{=}4n+2$ in order to have an equal number
of the different bond types.

We first demonstrate the presence of edge states in the SPT phases. For the
A$\Gamma$ phase, results for the 16 lowest states with the regular OBC at
$\phi\mathord{=}0.85\pi$ are obtained using ED (see Supplementary Figure 7).
Four low-lying states below the gap are clearly present. With increasing $N$,
these 4 states quickly become degenerate while the gap stabilizes at a finite
value. Similar results can be obtained for the A$K$ and F$K$ phase using the
slanted cluster, and a degeneracy of 4 is also observed for this case (see
Supplementary Figure 8). For the SPTα\- and SPTβ-phases, it has not been
possible to produce reliable results in the same manner, likely due to the
significantly larger correlation lengths.

Figure 8: Eigenvalues of total spin in the A$K$, F$K$ and A$\Gamma$ phases.
The eigenvalues $s_{\alpha}$=$s_{a,b,c}$ of the total spin
$S_{\alpha}$=$\sum_{i}S^{\alpha}_{i}$ in the four degenerate ground-states, in
zero field. Results are from ED. a A$K$-phase at $\phi=-0.08\pi$ with
$N\mathord{=}14,18\ldots 30$ b F$K$-phase at $\phi=0.95\pi$ with
$N\mathord{=}14,18\ldots 30$, c A$\Gamma$-phase at $\phi\mathord{=}0.85\pi$
with $N\mathord{=}12,14,\ldots 30$ (filled symbols).

To understand the nature of the edge-states, let us explore how the A$K$,
F$K$, and A$\Gamma$ phases respond to an external magnetic field. An external
magnetic field introduces an additional term in the Hamiltonian of the form
$H^{\prime}\mathord{=}g_{L}\mu_{B}{\bf B}\cdot{\bf S}_{\mathrm{tot}}$, where
${\bf S}_{\mathrm{tot}}\mathord{=}\sum_{i}\mathbf{S}_{i}$, $g_{L}$ is the
Landé factor and $\mu_{B}$ the Bohr magneton. Following Ref. [38] we denote
the 4 states $\psi_{1},\psi_{2},\psi_{3}$ and $\psi_{4}$ and consider
$\mathbf{S}_{\mathrm{tot},\alpha}$ in this four-fold degenerate space by
defining

$S^{\gamma\beta}_{\mathrm{tot},\alpha}=\langle\psi_{\gamma}|S_{\mathrm{tot},\alpha}|\psi_{\beta}\rangle,\
\ \gamma,\beta=1,2,3,4.$ (25)

Here, the components of the total spin $S_{\mathrm{tot},\alpha}$ are usually
taken to be identical to $x,y,z$ but given the underlying honeycomb structure
we shall find it useful to instead consider $\alpha$=$\mathbf{\hat{a}}$,
$\mathbf{\hat{b}}$, $\mathbf{\hat{c}}$ corresponding to the three directions
$[$11-2$]$, $[$1-10$]$ and $[111]$ that correspond to the perpendicular and
parallel to the z-bond, and perpendicular to the plane of the honeycomb (or
n-leg brick-wall), respectively. One then finds that the eigenvalues of the
matrices $(S^{\gamma\beta}_{\mathrm{tot},\alpha})$ for four corresponding
degenerate states are simply given by $(s_{\alpha},-s_{\alpha},0,0)$.

Our ED results for $s_{\alpha}$ are shown in Fig. 8. For the A$K$ phase at
$\phi\mathord{=-}0.08\pi$, shown in Fig. 8(a), we find
$s_{a}\mathord{\sim}3/4$, $s_{b}\mathord{\sim}1/2$ and
$s_{c}\mathord{\simeq}0.85$. Similarly, for the F$K$-phase at
$\phi\mathord{=}0.95\pi$, shown in Fig. 8(b), we find $s_{a}$=1,
$s_{b}\mathord{\sim}2$ and $s_{c}$=3/4. In both cases, we expect some
variation in the values of $s_{\alpha}$ as $\phi$ is tuned. We note that for
both the F$K$ and A$\Gamma$ phase, the values of $s_{\alpha}$ quickly saturate
at a small, finite value as $N$ is increased. This is indicative of
excitations localized at the edges as opposed to an actual magnetically
ordered ground-state which should show $s_{\alpha}$ continually growing with
$N$. A calculation of $s_{\alpha}$ for the SPTα and SPTβ-phases do not yield
clear results for the range of $N$ available with ED, as discussed in
Supplementary Note 6. In a realistic experimental setting, the presence of
impurities will always create finite open segments of ladders, with a
resulting Curie-law behavior. The response to an applied magnetic field is in
that case highly anisotropic and the low temperature Curie-law response should
show a strong directional dependence [38, 41, 43] with $\chi_{a}(T)$,
$\chi_{b}(T)$ and $\chi_{c}(T)$ clearly distinguishable. The results for the
A$\Gamma$-phase, shown in Fig. 8(c), are even more intriguing. They are not
only more anisotropic, but only $s_{b}$ is non-zero, approaching a value close
to 2 at $\phi\mathord{=}0.85\pi$. At a slightly different point in the
A$\Gamma$-phase with $\phi\mathord{=}0.8\pi$ we instead find
$s_{b}\mathord{\simeq}4/3$ but again only $s_{b}$ is non-zero. This implies
that the phase does not respond at all to a field applied along the
$\mathbf{\hat{a}}$ and $\mathbf{\hat{c}}$ directions, effectively
$g_{a},g_{c}\mathord{\simeq}0$.

Figure 9: Ground state magnetization in an infinitesimal field in the
A$\Gamma$ phase. ED results with $N$=24 for $\langle S^{a,b,c}_{i}\rangle$ in
the lowest state in the AG phase at $\phi/\pi$=0.85 with a field in the
$\mathbf{\hat{b}}$-direction of strength 0.002. The green, blue and cyan
colors indicate positive values while orange, red, and pink indicate negative
values. The size of the circles are proportional to the value of $\langle
S^{a,b,c}_{i}\rangle$. a $\langle S^{a}_{i}\rangle$, b $\langle
S^{b}_{i}\rangle$, c $\langle S^{c}_{i}\rangle$. Figure 10: The Schmidt gap in
the A$K$, F$K$ and A$\Gamma$ phases. The splitting of the entanglement
spectrum $\Delta\mathord{=}\lambda_{1}\mathord{-}\lambda_{2}$ from iDMRG
calculations with a field in the $\mathbf{\hat{a}}$, $\mathbf{\hat{b}}$ and
$\mathbf{\hat{c}}$ directions. a A$K$-phase at $\phi=-0.08\pi$ using
$\rho_{B}$. b F$K$-phase at $\phi=0.95\pi$ using $\rho_{B}$. c A$\Gamma$-phase
at $\phi\mathord{=}0.85\pi$ using $\rho_{A}$.

To gain a clearer picture of how the ladder in the A$\Gamma$ phase responds to
a magnetic field applied along the $\mathbf{\hat{b}}$-direction, we have
performed ED calculations in the presence of a small field in the
$\mathbf{\hat{b}}$-direction. The resulting site dependent magnetization
$\langle S^{a,b,c}_{i}\rangle$ can then easily be obtained for the
$\mathbf{\hat{a}},\mathbf{\hat{b}}$ and $\mathbf{\hat{c}}$-directions. Our
results are shown in Fig. 9 for $N$=24 as obtained from the ground-state with
a small field in the $\mathbf{\hat{b}}$-direction of 0.002 at $\phi/\pi$=0.85
in the A$\Gamma$-phase. The green, blue and cyan colors represent positive
expectation values, whereas orange, red and pink colors indicate negative
values, with the size of the points proportional to the expectation value. The
response along the $\mathbf{\hat{a}}$, $\mathbf{\hat{c}}$ directions is
completely symmetric in the positive and negative directions, yielding
$\sum_{i}S^{a,c}_{i}$=0. However, along the $\mathbf{\hat{b}}$-direction the
response is much larger, and we clearly find $\sum_{i}S^{b}_{i}\neq$0
consistent with the results shown in the SI (See Fig. S9) (c). With the field
along the $\mathbf{\hat{b}}$-direction, sizeable excitations are visible at
both ends of the ladder.

The response of the edge states to a magnetic field correlates with the
lifting (or absence of lifting) of the degeneracy in the entanglement
spectrum. As previously discussed, non-trivial indices in the projective
analysis can only arise if the degeneracy of all states in the entanglement
spectrum (ES) is larger than one. This implies that if a finite strength of
the perturbation is needed to remove the degeneracy, then a phase transition
does not occur until that strength is reached and the phase persists till that
point. On the other hand, if the degeneracy is lifted for any non-zero
strength of the perturbation, the symmetry protection is broken without an
associated phase transition. We can then investigate the response of the
degeneracy of the ES to a magnetic field in the $\mathbf{\hat{a}}$,
$\mathbf{\hat{b}}$ and $\mathbf{\hat{c}}$ directions. This is shown in Fig. 10
for the A$K$, F$K$, and A$\Gamma$ phases. For simplicity, we focus exclusively
on the difference in the two largest eigenvalues
$\Delta\mathord{=}\lambda_{1}\mathord{-}\lambda_{2}$, and in each case we
employ the reduced density matrix that has a two-fold degeneracy at zero
field.

As can be seen in Fig. 10(a) the degeneracy in the ES for the A$K$ phase is
immediately lifted by a field in any of the three directions which correlates
with the response of the edge-states (See Fig. S9(a) in the SI). However, the
response is rather weak, and a relatively large field has to be applied to see
a significant splitting. For the F$K$ phase, we have a similar effect as shown
in Fig. 10(b), but in this case the response is much stronger. However, for
the A$\Gamma$ phase, where we show results in Fig. 10(c), it is clear that a
field in the $\mathbf{\hat{a}}$ direction of around $h_{a}$=0.06 is needed to
lift the degeneracy of the ES. For a field applied in the $\mathbf{\hat{c}}$
direction, a significantly larger field is needed. On the other hand, a field
in the $\mathbf{\hat{b}}$ direction immediately induces a large splitting in
the entanglement spectrum even for infinitesimal field strengths. Since the
degeneracy remains intact in the $\mathbf{\hat{a}}$ and $\mathbf{\hat{c}}$
directions, we conclude that the SPT character of the A$\Gamma$ phase persists
with respect to a field applied in the $\mathbf{\hat{a}}$ and
$\mathbf{\hat{c}}$ directions.

The A$\Gamma$ phase is then protected by the product of time-reversal
($\mathrm{TR}$) and $\pi$ rotation around the $\mathbf{\hat{b}}$-axis
($\mathcal{R}_{b}$), $\mathrm{TR}\times\mathcal{R}_{b}$, the only remaining
symmetry [60] when the field is in the ${\mathbf{\hat{a}}}{\mathbf{\hat{c}}}$
plane, but broken when it is along the ${\hat{b}}$-axis. Hence, if a field is
applied in the ${\mathbf{\hat{a}}}{\mathbf{\hat{c}}}$ plane, a transition to
the trivial polarized state can only occur at finite field strengths with
potentially other phases intervening before the polarized state is
encountered. Several such transitions were observed for the A$\Gamma$ phase
(denoted by K$\Gamma$SL) for a field in the ${\mathbf{\hat{c}}}$ direction
[18, 30]. On the other hand, the F$K$ phase is not protected by the
$\mathrm{TR}\times\mathcal{R}_{b}$ symmetry, and if a field is applied in the
${\mathbf{\hat{c}}}$ direction the ES degeneracy is lost as shown in Fig.
10(b). However, the field induced F$K$ phase can still be distinguished from
the polarized state.

## Discussion

Our initial inquiry in this paper pertains to the nature of the A$\Gamma$
phase and whether there exists a defining quantity for its characterization.
For example, the Kitaev phases (A$K$ and F$K$) in the ladder display the
character of SPT phases. It is likely that A$\Gamma$ is another SPT phase. If
so, we expect all the signatures of the SPT such as the degeneracy of the
entanglement spectrum, ground state degeneracy under OBC, and the presence of
a SOP. Using the iDMRG, DMRG, and ED techniques, we indeed found that the
entanglement spectrum is degenerate and there exists four-fold ground state
degeneracy under the regular OBC in the A$\Gamma$ phase. It is interesting to
note that the same results were obtained for the Kitaev phases, A$K$ and F$K$,
but under the slanted OBC.

Despite such clear signatures of the SPT, determining the corresponding SOP in
the A$\Gamma$ phase has been challenging. We found that the string order
correlation function is related to ordinary local order in a regular
correlation function in a model $H_{\mathrm{KQ}}$ obtained only after two
consecutive unitary transformations. Hence, we term this order as ’twice’
hidden.

To understand the symmetry that protects the A$\Gamma$ phase, we also
investigated the effects of the external magnetic field. From the magnetic
field response, we noted that the A$\Gamma$ phase is completely inert to the
magnetic field when the field is applied in the ${\hat{a}}$ and ${\hat{c}}$
directions, which correspond to perpendicular to the z-bond and the ladder
plane, respectively. Accordingly, the entanglement spectrum degeneracy remains
intact when the field is applied in the ${\hat{a}}$ and ${\hat{c}}$
directions. This is in contrast to the effect of applying the magnetic field
along the ${\hat{b}}$ direction, i.e., parallel to the z-bond, which
immediately lifts the degeneracy of the entanglement spectrum. We note that
the product of $\mathrm{TR}$ and $\mathcal{R}_{b}$,
$\mathrm{TR}\times\mathcal{R}_{b}$ symmetry, is preserved when the magnetic
field is applied in the ${\mathbf{\hat{a}}}{\mathbf{\hat{c}}}$-plane, which is
valid for the generic honeycomb Kitaev model beyond the ladder[60]. Thus, we
conclude that the A$\Gamma$ is protected by the
$\mathrm{TR}\times\mathcal{R}_{b}$ symmetry. Another intriguing implication
from the magnetic field study is that the edge state in the A$\Gamma$ phase is
not a isotropic free spin-1/2 unlike the standard S=1 Haldane SPT. They act
like spinless modes under the field in ${\hat{a}}$ and ${\hat{c}}$ directions.
Further studies are needed to fully understand the nature of the zero-energy
modes at the boundary of the system with the regular OBC.

In the context of Kitaev materials, let us revisit our motivations for
investigating the A$\Gamma$ phase in the ladder model. As previously mentioned
in the introduction, the majority of $d^{5}$ Kitaev materials prominently
feature FM Kitaev and AFM $\Gamma$ interactions. However, ongoing debates
persist regarding the specific phase that arises in this region. Several
numerical studies have suggested the presence of magnetic disorder [12, 13,
14, 15, 16, 17, 18, 19], while others have indicated a magnetically ordered
phase, such as a zig-zag order [21]. Should a zig-zag order indeed be manifest
in the 2D limit, we would anticipate observing the same ordering pattern in
the 2-leg ladder, as the magnetic unit cell of the zig-zag can be captured in
the ladder geometry. This is indeed the case for the Kitaev-Heisenberg ladder
model, where the zig-zag, stripy, and FM ordered phases reported in the 2D
honeycomb clusters are found in the ladder geometry [23].

Our findings have substantiated the presence of disordered state in the
A$\Gamma$ phase of the 2-leg ladder, categorizing it as a SPT phase
characterized by a SOP. The 2D limit can be constructed by stacking the
ladders, and one possibility of the resulting 2D phase is a stacked SPTs with
edge modes known as a weak-SPT [61]. However, the coupling between the ladders
may generate a new phase or critical point. It is interesting to note that the
evolution from a stacked weak-SPT chains to a gapless critical point
supporting edge modes that do not hybridize with bulk modes was reported in
the extended anisotropic Kitaev model approaching from the dimer limit [62].
Our findings hint at the possibility that as the 2D limit is approached, the
A$\Gamma$ phase may become a 2D spin liquid, denoted as the K$\Gamma$ spin
liquid. However, we cannot rule out a possibility of large unit cell [63] or
incommensurate [20] magnetic orders whose magnetic unit cells are beyond the
ladder geometry, and a definitive resolution to this question remains a
subject for future investigation.

## Methods

###  .1 Numerical Methods

We use a fully parallelized implementation [64] of the Lanczos algorithm to
perform the exact diagonalizations (ED) of ladders with up to $N$=30 using
both open and periodic boundary conditions. In addition to exact
diagonalizations, we use finite size density matrix renormalization group [65,
66, 67, 68, 69, 70] (DMRG) to study both the K$\Gamma$ model, Eq. (1) and
$H_{\mathrm{K}\Gamma}^{U_{6}}$ under both periodic (PBC) and open (OBC)
boundary conditions, with the main part of our results obtained from the
infinite DMRG [71, 70] (iDMRG) variant of DMRG. The iDMRG calculations were
performed with unit cells of 12, 24 or 60 sites. Typical precision for both
DMRG and iDMRG are $\epsilon<10^{-11}$ with a bond dimension in excess of
1000.

###  .2 Energy susceptibility

To determine the phase diagram we study the susceptibility derived from the
ground-state energy per spin, $e_{0}$:

$\chi_{\phi}^{e}=-\frac{\partial^{2}e_{0}}{\partial\phi^{2}},\ $ (26)

At a quantum critical point (QCP) it is known [72] that, for a finite system
of size $N$, the energy susceptibility diverges as

$\chi^{e}\sim N^{2/\nu-(d+z)}.$ (27)

Here $\nu$ and $z$ are the correlation and dynamical critical exponents and
$d$ is the dimension. Hence, $\chi^{e}$ only diverges at the phase transition
if the critical exponent $\nu$ is smaller than $2/(d+z)$. For the present case
we have $d$=$1$ and if we assume $z$=1, then a divergence is observed only if
$\nu<1$.

###  .3 Entanglement Entropy and Spectrum, Schmidt gap

When studying the ladder shown in Fig. 1a it is important to realize that
there are different ways of partitioning the system in two partitions of size
$x$ and $N-x$. This is crucial when considering the bipartite von Neumann
entanglement entropy, $EE$, as well as for the entanglement spectrum [73] of
central importance for understanding topological properties [73, 34, 74, 75].
Both are obtained from the spectrum of the reduced density matrix, $\rho_{x}$,
of either one of the two partitions. Here we focus on two specific partitions
shown in Fig. 1a as the red and blue dashed lines. With the numbering in Fig.
1a, they correspond to either an odd ($N/2$-1, red) or even ($N/2$, blue)
number of sites in the partitions. We refer to the density matrix derived from
the former case with $N/2$-1 as $\rho_{B}$ and to the latter case with $N/2$
as $\rho_{A}$. We mainly focus on the case where the number of sites in the
partition is close to the mid-point, either $N/2-1$ ($\rho_{B}$) or $N/2$
($\rho_{A}$). but when considering the bipartite entanglement entropy, we let
the number in the partition vary but only consider an even number of sites in
the sub system corresponding to moving the blue dashed line in Fig. 1a along
the ladder. For a subsystem, $A$, of size $x$ the entanglement entropy is
defined by:

$EE(x)=-\mathrm{Tr}\rho_{x}\ln\rho_{x}.$ (28)

Our results for $EE(x)$ can be found in Supplementary Note 2. The eigenvalues,
$l_{\alpha}$, of the reduced density matrix, $\rho_{x}$, correspond to the
Schmidt decomposition, $l_{\alpha}$=$\lambda_{\alpha}^{2}$ and thereby the
entanglement spectrum [73], which then will depend on whether $\rho_{A}$ or
$\rho_{B}$ from Fig. 1a is used. The Schmidt gap is then defined as
$\lambda_{1}-\lambda_{2}$.

## Data Availability

The data that support the findings of this study are available at
10.5281/zenodo.10443031, or alternatively from the corresponding authors upon
reasonable request.

## Code Availability

The code used to generate the data used in this study is available from the
corresponding author upon reasonable request.

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###### Acknowledgements.

We thank A. Catuneanu and J. Gordon for fruitful discussions during the
initial stages of this work. We acknowledge the support of the Natural
Sciences and Engineering Research Council of Canada (NSERC) through Discovery
Grants (No. RGPIN-2017-05759 and No. RGPIN-2022-04601). H.-Y.K. acknowledges
the support of CIFAR and the Canada Research Chairs Program. This research was
enabled in part by support provided by SHARCNET (sharcnet.ca) and the Digital
Research Alliance of Canada (alliancecan.ca). Part of the numerical
calculations were performed using the ITensor library [76].

## Author contributions

Density-matrix renormalization group and exact diagonalization calculations
were performed by E.S.S. H.-Y. K. initiated the project. All authors managed
the project and wrote the manuscript.

## Competing Interests

The authors declare no Competing Financial or Non-Financial Interests.

See pages 1, of SI.pdf See pages 0, of SI.pdf