5/7 Skewed Ladder Quantum Phases

• The 5/7 skewed ladder is a frustrated two-leg quantum spin model with alternating five- and seven-membered rings that induce geometric frustration and diverse quantum phases.
• It employs the isotropic Heisenberg Hamiltonian along with entanglement entropy and fidelity measures to pinpoint quantum phase transitions and reentrant singlet regimes.
• The model reveals distinctive magnetic behaviors, including nonmagnetic, ferrimagnetic phases, and quantized magnetization plateaus under applied magnetic fields.

Searching arXiv for papers on the 5/7 skewed ladder and closely related skewed-ladder quantum phases. The 5/7 skewed ladder is a frustrated two-leg quantum spin ladder whose unit cell is built from alternately fused five-membered and seven-membered rings. In the condensed-matter literature, it is treated as the spin-lattice analog of a fused azulene chain and as a ladder obtained from a zigzag ladder by periodically removing or shifting rung bonds, so that the rungs do not form a regular vertical pattern. This skewed connectivity generates geometric frustration because the antiferromagnetic exchanges cannot all be simultaneously satisfied. For isotropic antiferromagnetic Heisenberg models on this geometry, the 5/7 ladder exhibits nonmagnetic, magnetic, ferrimagnetic, reentrant singlet, and symmetry-broken regimes, with quantum phase transitions diagnosed by spin gaps, entanglement entropy, fidelity, and magnetization plateaus (Das et al., 2023, Das et al., 2022, Dey et al., 2020).

1. Geometry and defining structural features

The defining feature of the 5/7 skewed ladder is its alternating five- and seven-membered-ring topology. In geometric language, it is a two-leg ladder with a skewed rung pattern: some nearest-neighbor couplings are slanted rather than vertical, and the lattice can be viewed as a ladder with periodically missing or shifted bonds. In the standard unit-cell description used for the spin models, the ladder has 8 spins and 10 bonds per unit cell (Das et al., 2023, Dey et al., 2020).

This geometry is the source of the model’s frustration. The exchange network contains two rung bonds and a larger set of leg and diagonal couplings, and the resulting odd-membered loops prevent simultaneous minimization of all antiferromagnetic bonds. In the language used in the review literature, the 5/7 ladder is one of the “skewed spin ladders” that display “completely different behaviour” from other skewed geometries such as the 3/4 and 3/5 systems when the Hamiltonian parameter is varied (Das et al., 2023).

A recurrent structural interpretation is that the lattice is composed of local motifs that can support both strong singlet formation and effectively unpaired spins. This underlies the coexistence of dimer-like physics, progressively higher-spin ground states, and reentrant nonmagnetic behavior reported across the spin-$\tfrac12$ and spin-1 studies (Dey et al., 2020, Das et al., 2020).

2. Heisenberg Hamiltonian and control parameter

For the isotropic spin-$\tfrac12$ Heisenberg model, the 5/7 ladder Hamiltonian is written as

$$H_{5/7} = J_1 \sum_i \left(\vec{S}_{i,1}\cdot \vec{S}_{i,2} + \vec{S}_{i,4}\cdot \vec{S}_{i,5}\right) + J_2 \sum_i \bigg( \vec{S}_{i,7}\cdot \vec{S}_{i+1,1} + \vec{S}_{i,8}\cdot \vec{S}_{i+1,2} + \sum_{k=1}^{6}\vec{S}_{i,k}\cdot \vec{S}_{i,k+2} \bigg).$$

Here $J_1$ is the exchange on the two rung bonds in each unit cell, while $J_2$ acts on the remaining leg and diagonal bonds that generate frustration. All exchanges are antiferromagnetic, and the standard convention is to fix $J_2=1$ as the energy scale, so the tuning parameter is the ratio $J_1/J_2$, usually written simply as $J_1$ (Das et al., 2022, Das et al., 2023).

The same exchange topology is also used in the spin-1 formulation, with the spin operators promoted to spin-1 variables: $$H_{5/7} = J_1 \sum_i \left( \vec S_{i,1}\cdot \vec S_{i,2} + \vec S_{i,4}\cdot \vec S_{i,5} \right) + J_2 \sum_i \left( \vec S_{i,7}\cdot \vec S_{i+1,1} + \vec S_{i,8}\cdot \vec S_{i+1,2} + \sum_{k=1}^{6} \vec S_{i,k}\cdot \vec S_{i,k+2} \right),$$ again with $J_2=1$ and varying $\tfrac12$0 (Das et al., 2020).

In a magnetic field, the spin-$\tfrac12$1 model includes a Zeeman term,

$\tfrac12$2

and the field-dependent levels satisfy

$\tfrac12$3

This makes plateau formation a level-crossing problem between neighboring $\tfrac12$4 sectors (Dey et al., 2020, Das et al., 2023).

3. Zero-field phase structure for the spin-$\tfrac12$5 ladder

At zero field, the spin-$\tfrac12$6 5/7 ladder displays a sequence of ground-state changes as $\tfrac12$7 increases. The broad picture given in the review is that the ladder starts from a singlet ground state at small $\tfrac12$8, enters progressively higher-spin states, then shows a reentrant singlet phase, and finally reaches the highest-spin state

$\tfrac12$9

where $$H_{5/7} = J_1 \sum_i \left(\vec{S}_{i,1}\cdot \vec{S}_{i,2} + \vec{S}_{i,4}\cdot \vec{S}_{i,5}\right) + J_2 \sum_i \bigg( \vec{S}_{i,7}\cdot \vec{S}_{i+1,1} + \vec{S}_{i,8}\cdot \vec{S}_{i+1,2} + \sum_{k=1}^{6}\vec{S}_{i,k}\cdot \vec{S}_{i,k+2} \bigg).$$0 is the number of unit cells (Das et al., 2023).

For the 24-site periodic system studied in the entanglement-and-fidelity analysis, the phase structure is resolved more explicitly. The ground state changes from a singlet to a triplet at

$$H_{5/7} = J_1 \sum_i \left(\vec{S}_{i,1}\cdot \vec{S}_{i,2} + \vec{S}_{i,4}\cdot \vec{S}_{i,5}\right) + J_2 \sum_i \bigg( \vec{S}_{i,7}\cdot \vec{S}_{i+1,1} + \vec{S}_{i,8}\cdot \vec{S}_{i+1,2} + \sum_{k=1}^{6}\vec{S}_{i,k}\cdot \vec{S}_{i,k+2} \bigg).$$1

becomes singlet again for

$$H_{5/7} = J_1 \sum_i \left(\vec{S}_{i,1}\cdot \vec{S}_{i,2} + \vec{S}_{i,4}\cdot \vec{S}_{i,5}\right) + J_2 \sum_i \bigg( \vec{S}_{i,7}\cdot \vec{S}_{i+1,1} + \vec{S}_{i,8}\cdot \vec{S}_{i+1,2} + \sum_{k=1}^{6}\vec{S}_{i,k}\cdot \vec{S}_{i,k+2} \bigg).$$2

and reaches

$$H_{5/7} = J_1 \sum_i \left(\vec{S}_{i,1}\cdot \vec{S}_{i,2} + \vec{S}_{i,4}\cdot \vec{S}_{i,5}\right) + J_2 \sum_i \bigg( \vec{S}_{i,7}\cdot \vec{S}_{i+1,1} + \vec{S}_{i,8}\cdot \vec{S}_{i+1,2} + \sum_{k=1}^{6}\vec{S}_{i,k}\cdot \vec{S}_{i,k+2} \bigg).$$3

The same study emphasizes that the system is nonmagnetic at small $$H_{5/7} = J_1 \sum_i \left(\vec{S}_{i,1}\cdot \vec{S}_{i,2} + \vec{S}_{i,4}\cdot \vec{S}_{i,5}\right) + J_2 \sum_i \bigg( \vec{S}_{i,7}\cdot \vec{S}_{i+1,1} + \vec{S}_{i,8}\cdot \vec{S}_{i+1,2} + \sum_{k=1}^{6}\vec{S}_{i,k}\cdot \vec{S}_{i,k+2} \bigg).$$4, magnetic at intermediate $$H_{5/7} = J_1 \sum_i \left(\vec{S}_{i,1}\cdot \vec{S}_{i,2} + \vec{S}_{i,4}\cdot \vec{S}_{i,5}\right) + J_2 \sum_i \bigg( \vec{S}_{i,7}\cdot \vec{S}_{i+1,1} + \vec{S}_{i,8}\cdot \vec{S}_{i+1,2} + \sum_{k=1}^{6}\vec{S}_{i,k}\cdot \vec{S}_{i,k+2} \bigg).$$5, reenters a nonmagnetic region, and then becomes magnetic again (Das et al., 2022).

The review article presents a closely related finite-size description: for fixed system size, $$H_{5/7} = J_1 \sum_i \left(\vec{S}_{i,1}\cdot \vec{S}_{i,2} + \vec{S}_{i,4}\cdot \vec{S}_{i,5}\right) + J_2 \sum_i \bigg( \vec{S}_{i,7}\cdot \vec{S}_{i+1,1} + \vec{S}_{i,8}\cdot \vec{S}_{i+1,2} + \sum_{k=1}^{6}\vec{S}_{i,k}\cdot \vec{S}_{i,k+2} \bigg).$$6 increases with $$H_{5/7} = J_1 \sum_i \left(\vec{S}_{i,1}\cdot \vec{S}_{i,2} + \vec{S}_{i,4}\cdot \vec{S}_{i,5}\right) + J_2 \sum_i \bigg( \vec{S}_{i,7}\cdot \vec{S}_{i+1,1} + \vec{S}_{i,8}\cdot \vec{S}_{i+1,2} + \sum_{k=1}^{6}\vec{S}_{i,k}\cdot \vec{S}_{i,k+2} \bigg).$$7 up to about $$H_{5/7} = J_1 \sum_i \left(\vec{S}_{i,1}\cdot \vec{S}_{i,2} + \vec{S}_{i,4}\cdot \vec{S}_{i,5}\right) + J_2 \sum_i \bigg( \vec{S}_{i,7}\cdot \vec{S}_{i+1,1} + \vec{S}_{i,8}\cdot \vec{S}_{i+1,2} + \sum_{k=1}^{6}\vec{S}_{i,k}\cdot \vec{S}_{i,k+2} \bigg).$$8, then the system reenters a singlet phase in the window

$$H_{5/7} = J_1 \sum_i \left(\vec{S}_{i,1}\cdot \vec{S}_{i,2} + \vec{S}_{i,4}\cdot \vec{S}_{i,5}\right) + J_2 \sum_i \bigg( \vec{S}_{i,7}\cdot \vec{S}_{i+1,1} + \vec{S}_{i,8}\cdot \vec{S}_{i+1,2} + \sum_{k=1}^{6}\vec{S}_{i,k}\cdot \vec{S}_{i,k+2} \bigg).$$9

and for larger systems the increase in $J_1$0 becomes saturated once

$J_1$1

in the sense that $J_1$2 for $J_1$3. In that strong-$J_1$4 limit, each unit cell contributes two unpaired spins to the total ground-state spin (Das et al., 2023).

The physical interpretation given in the review is that the initial singlet phase is a nonmagnetic antiferromagnetic state dominated by quantum singlets, the higher-spin regions reflect frustration-induced uncompensated moments, and the reentrant singlet regime is associated with long-wavelength spin-density-wave correlations rather than simple local dimerization (Das et al., 2023). This suggests that the 5/7 ladder should be understood not as a single crossover problem but as a sequence of distinct quantum phases selected by competition between rung singlets, frustrated leg couplings, and symmetry-sector rearrangements.

4. Entanglement entropy, fidelity, and symmetry-resolved transitions

A distinctive feature of the 5/7 ladder literature is the use of entanglement entropy (EE) and fidelity as precision diagnostics of quantum phase transitions. In the 24-site periodic study, the system is divided into equal halves and the reduced density matrix of one subsystem is defined from

$J_1$5

with matrix elements

$J_1$6

The von Neumann entropy is then

$J_1$7

and the fidelity between nearby couplings is

$J_1$8

with fidelity susceptibility

$J_1$9

For the 5/7 ladder, the main reported signatures are that EE shows a discontinuous jump while fidelity shows a sharp dip at the transition points (Das et al., 2022).

The first major transition occurs at $J_2$0, where both quantities change abruptly, identifying the singlet-to-triplet transition. Additional sharp structures occur around

$J_2$1

In particular, the fidelity dip at $J_2$2 corresponds to a maximum in the symmetry gap $J_2$3, while at $J_2$4 the gap vanishes and the system enters a degenerate region. For $J_2$5, both EE and fidelity become smooth or nearly constant again, consistent with a stable phase (Das et al., 2022).

A central technical issue is degeneracy across symmetry subspaces. The 5/7 ladder has a reflection symmetry perpendicular to the legs, and the states are classified into $J_2$6 and $J_2$7 sectors. In regions where the lowest states in different symmetry sectors are degenerate, unsymmetrized numerical diagonalization can select different linear combinations at neighboring parameter values, causing violent but unphysical oscillations in EE and fidelity. The symmetry-resolved analysis removes this artifact.

$J_2$8 range	lowest-state symmetry	interpretation
$J_2$9 and $J_2=1$0	$J_2=1$1	nondegenerate symmetry sector
$J_2=1$2 and $J_2=1$3	$J_2=1$4	nondegenerate symmetry sector
$J_2=1$5 and $J_2=1$6	doubly degenerate	$J_2=1$7; unsymmetrized EE and fidelity are unreliable

The corresponding methodological conclusion is explicit: when the ground state is degenerate, unsymmetrized EE and fidelity can fluctuate wildly even without an actual phase transition, whereas calculations performed in the lowest-energy state of each symmetry subspace suppress these fluctuations and isolate the true phase boundaries (Das et al., 2022). A frequent misconception is therefore that every sharp oscillation in unsymmetrized diagnostics indicates an additional phase transition; in the 5/7 ladder, the symmetry-resolved analysis shows that many such features are instead artifacts of degeneracy.

5. Magnetic field response and magnetization plateaus

In a Zeeman field, the spin-$J_2=1$8 5/7 ladder shows a characteristic plateau sequence at

$J_2=1$9

where $J_1/J_2$0 is the normalized magnetization. These plateaus are consistent with the Oshikawa–Yamanaka–Affleck condition. For the 5/7 unit cell with $J_1/J_2$1 spins of size $J_1/J_2$2, the condition

$J_1/J_2$3

becomes

$J_1/J_2$4

which allows

$J_1/J_2$5

The nontrivial observed plateaus are $J_1/J_2$6 (Dey et al., 2020, Das et al., 2023).

The strong-coupling interpretation is particularly clear. In the large-$J_1/J_2$7 regime, the zero-field ground state can be approximated by three weakly coupled singlet dimers and two free spins per unit cell. For

$J_1/J_2$8

the thermodynamic ground state already has

$J_1/J_2$9

at zero field. As the field is increased, the singlets become triplets progressively, producing the sequence

$J_1$0

This directly yields the $J_1$1, $J_1$2, and $J_1$3 plateaus (Dey et al., 2020).

The perturbative strong-$J_1$4 treatment gives explicit second-order energies for the relevant plateau states. For example,

$J_1$5

for the $J_1$6 state,

$J_1$7

for the $J_1$8 state, and

$J_1$9

for the $$H_{5/7} = J_1 \sum_i \left( \vec S_{i,1}\cdot \vec S_{i,2} + \vec S_{i,4}\cdot \vec S_{i,5} \right) + J_2 \sum_i \left( \vec S_{i,7}\cdot \vec S_{i+1,1} + \vec S_{i,8}\cdot \vec S_{i+1,2} + \sum_{k=1}^{6} \vec S_{i,k}\cdot \vec S_{i,k+2} \right),$$0 state. Equating neighboring energies gives the critical fields

$$H_{5/7} = J_1 \sum_i \left( \vec S_{i,1}\cdot \vec S_{i,2} + \vec S_{i,4}\cdot \vec S_{i,5} \right) + J_2 \sum_i \left( \vec S_{i,7}\cdot \vec S_{i+1,1} + \vec S_{i,8}\cdot \vec S_{i+1,2} + \sum_{k=1}^{6} \vec S_{i,k}\cdot \vec S_{i,k+2} \right),$$1

$$H_{5/7} = J_1 \sum_i \left( \vec S_{i,1}\cdot \vec S_{i,2} + \vec S_{i,4}\cdot \vec S_{i,5} \right) + J_2 \sum_i \left( \vec S_{i,7}\cdot \vec S_{i+1,1} + \vec S_{i,8}\cdot \vec S_{i+1,2} + \sum_{k=1}^{6} \vec S_{i,k}\cdot \vec S_{i,k+2} \right),$$2

$$H_{5/7} = J_1 \sum_i \left( \vec S_{i,1}\cdot \vec S_{i,2} + \vec S_{i,4}\cdot \vec S_{i,5} \right) + J_2 \sum_i \left( \vec S_{i,7}\cdot \vec S_{i+1,1} + \vec S_{i,8}\cdot \vec S_{i+1,2} + \sum_{k=1}^{6} \vec S_{i,k}\cdot \vec S_{i,k+2} \right),$$3

The perturbation theory is reported to agree well with DMRG and ED for $$H_{5/7} = J_1 \sum_i \left( \vec S_{i,1}\cdot \vec S_{i,2} + \vec S_{i,4}\cdot \vec S_{i,5} \right) + J_2 \sum_i \left( \vec S_{i,7}\cdot \vec S_{i+1,1} + \vec S_{i,8}\cdot \vec S_{i+1,2} + \sum_{k=1}^{6} \vec S_{i,k}\cdot \vec S_{i,k+2} \right),$$4 (Dey et al., 2020).

Finite-temperature calculations show that the $$H_{5/7} = J_1 \sum_i \left( \vec S_{i,1}\cdot \vec S_{i,2} + \vec S_{i,4}\cdot \vec S_{i,5} \right) + J_2 \sum_i \left( \vec S_{i,7}\cdot \vec S_{i+1,1} + \vec S_{i,8}\cdot \vec S_{i+1,2} + \sum_{k=1}^{6} \vec S_{i,k}\cdot \vec S_{i,k+2} \right),$$5 and $$H_{5/7} = J_1 \sum_i \left( \vec S_{i,1}\cdot \vec S_{i,2} + \vec S_{i,4}\cdot \vec S_{i,5} \right) + J_2 \sum_i \left( \vec S_{i,7}\cdot \vec S_{i+1,1} + \vec S_{i,8}\cdot \vec S_{i+1,2} + \sum_{k=1}^{6} \vec S_{i,k}\cdot \vec S_{i,k+2} \right),$$6 plateaus are robust, whereas the $$H_{5/7} = J_1 \sum_i \left( \vec S_{i,1}\cdot \vec S_{i,2} + \vec S_{i,4}\cdot \vec S_{i,5} \right) + J_2 \sum_i \left( \vec S_{i,7}\cdot \vec S_{i+1,1} + \vec S_{i,8}\cdot \vec S_{i+1,2} + \sum_{k=1}^{6} \vec S_{i,k}\cdot \vec S_{i,k+2} \right),$$7 plateau shrinks rapidly and survives only up to about

$$H_{5/7} = J_1 \sum_i \left( \vec S_{i,1}\cdot \vec S_{i,2} + \vec S_{i,4}\cdot \vec S_{i,5} \right) + J_2 \sum_i \left( \vec S_{i,7}\cdot \vec S_{i+1,1} + \vec S_{i,8}\cdot \vec S_{i+1,2} + \sum_{k=1}^{6} \vec S_{i,k}\cdot \vec S_{i,k+2} \right),$$8

The reported reason is that the $$H_{5/7} = J_1 \sum_i \left( \vec S_{i,1}\cdot \vec S_{i,2} + \vec S_{i,4}\cdot \vec S_{i,5} \right) + J_2 \sum_i \left( \vec S_{i,7}\cdot \vec S_{i+1,1} + \vec S_{i,8}\cdot \vec S_{i+1,2} + \sum_{k=1}^{6} \vec S_{i,k}\cdot \vec S_{i,k+2} \right),$$9 plateau has the smallest zero-temperature width and the smallest excitation gap to neighboring magnetization sectors. At plateau edges, the magnetization cusps follow the algebraic square-root form

$J_2=1$0

which is the expected behavior for 1D gapped-to-gapless transitions (Dey et al., 2020).

6. Symmetry breaking, spin-1 extension, and broader significance

The 5/7 skewed ladder also supports a rich symmetry structure. In the spin-$J_2=1$1 review, the ladder is described as having mirror symmetry $J_2=1$2 and spin inversion symmetry $J_2=1$3. Degeneracy between the lowest states in the even and odd mirror sectors signals broken mirror symmetry and is identified with bond-order-wave behavior. Degeneracy between the $J_2=1$4 and $J_2=1$5 spin-inversion sectors signals broken spin inversion symmetry and is identified with spin-density-wave order. For the 5/7 ladder, the singlet region around

$J_2=1$6

shows mirror-symmetry breaking in finite systems, while the interval

$J_2=1$7

is associated with a triplet ground state and broken mirror symmetry, identified by the authors with vector chiral symmetry breaking without external magnetic field or anisotropic exchange (Das et al., 2023).

The spin-1 5/7 ladder preserves the same geometric exchange pattern but displays a different phase diagram. The reported phases are: a nonmagnetic AF phase for

$J_2=1$8

a ferrimagnetic phase with

$J_2=1$9

a reentrant nonmagnetic phase for

$\tfrac12$00

and a higher ferrimagnetic phase with

$\tfrac12$01

The same study reports spin-current-carrying points near

$\tfrac12$02

arising from simultaneous breaking of reflection and spin parity symmetries (Das et al., 2020).

Across the skewed-ladder family, the 5/7 geometry is therefore a reference case for geometry-driven frustration. The review contrasts it with the 3/4 ladder, which goes from a singlet to a high-spin state with each unit cell contributing spin $\tfrac12$03, and with the 3/5 ladder, which shows a singlet phase at small parameter values, a high-spin regime at intermediate values, and a reentrant singlet at still higher values (Das et al., 2023). A plausible implication is that the 5/7 ladder is especially useful for isolating which features arise specifically from the alternating five- and seven-ring topology rather than from skewed-ladder frustration in general.

The broader significance of the 5/7 skewed ladder is not limited to abstract model building. The geometry has been connected to fused azulene-like carbon structures and has been proposed as relevant to graphene grain boundaries or defect networks (Dey et al., 2020). Within that context, the 5/7 ladder serves as a concrete example of how fused-ring topology, antiferromagnetic exchange, and quantum fluctuations can combine to produce reentrant nonmagnetism, ferrimagnetism, symmetry breaking, and quantized magnetization plateaus in a single low-dimensional system.