Multipole phases in a type of spin/fermion ladders with local conserved quantities and generalizations

TL;DR

The paper develops exactly solvable multipole phases on spin ladders by introducing local dimer conserved quantities $\tau_n$, which fragment the Hilbert space into sectors described by a first-order spin $S_n$ atop the original spins $\boldsymbol{\sigma}$. By mapping each sector to a transverse-field Ising model in a static $\tau$ background, it identifies dipole, charge-pair, and higher-order multipole phases, with phase structure governed by $\tau$ configurations and effective-spin interactions; the dynamical structure factor $D^{zz}$ exposes clear experimental fingerprints. The authors provide explicit solvable vertical and horizontal ladder geometries with quadratic couplings, derive phase diagrams via exact fermionic or spin-diagonalization methods, and show how $D^{zz}$ distinguishes phases. They further generalize to quadrupole and octopole models by introducing additional dimer conservations, illustrating a hierarchical mapping to successive Ising-like descriptions and suggesting potential realizations in materials with dimer or plaquette interactions, along with directions for stability analysis and extension to higher dimensions.

Abstract

We study spin/fermion ladder models with exact multipole phases, which are traditional spin phases formed by multipole moments. These phases feature non-trivial order with zero magnetization. The multipole models have dimer local conserved quantities that are Ising terms of spin. The Hilbert spaces are locally fragmented into independent sectors described effectively by {\it higher-order spin}. For dipole models, we consider two ladder geometries with quadratic spin couplings and work out the phase diagrams. Different phases of the model can be distinguished experimentally with the dynamical structure factor. Higher-order multipole models are obtained by introducing more dimer conserved quantities. The phases are characterized by the values of the local conserved quantities and the traditional spin phases of the higher-order spin.

Figures (7)

• Figure 1: Two ladder geometries for dipole models with quadratic spin couplings. The numbering of the sites marks the zig-zag string, along which the Jordan-Wigner transformations are performed. The $J$-bonds are represented by green solid lines while $K$-bonds are black lines (solid lines for coupling constant $K$ and dashed for $\tilde{K}$).
• Figure 2: Phases of the vertical ladder. Upper: illustration of different phases with arrows representing the first-order spin $S^{z}$, filled (empty) circles representing up (down) spins. Lower: phase diagram of the vertical ladder (quadratic spin coupling) with $\tilde{K}=0.8$, $J_{x}=J_{y}=1$ obtained from the fermionic ground-state energy in \ref{['spinladderH']}.
• Figure 3: Phases of the horizontal ladder. Upper: schematic phase diagram of the ANNNI chain which we adopt in this work. Lower: phase diagram of the horizontal ladder with $K=-4$, $J_{x}=J_{y}=-1$ (corresponding to the dashed line in the upper figure).
• Figure 4: The quadrupole ladder with zeroth-order spin $\sigma$. The plaquette terms are represented by the green squares; green lines mark the ZZ coupling that commute with the Hamiltonian. Introducing first-order spin $S$, the model becomes a dipole chain which is further brought down to transverse-field Ising model with second-order spin $\mathbb{S}$.
• Figure S1: Two ladder geometries considered in this work presented in a different way. The numbering of the sites marks the zig-zag string, along which the Jordan-Wigner transformations are performed. The $J$-bonds (XYZ coupling) are represented by green solid lines, $K$-bonds (ZZ coupling) are black lines, both solid and dashed lines are used to mark the two coupling constants $K$ and $\tilde{K}$.