Immobile and mobile excitations of three-spin interactions on the diamond chain

TL;DR

This work analyzes a solvable 1D spin-1/2 model on a diamond chain with three-spin interactions that supports both mobile excitations and fully immobile excitations protected by local subsystem symmetries. The authors derive an exact mapping to an array of independent transverse-field Ising chain segments, enabling analytic solutions for the mobile-mode dispersions and the immobile-mode energy via ground-state energy differences between symmetry sectors. They identify a second-order phase transition driven by mobile modes at $(J/h)_{c} = \sqrt{2}$, and they obtain a closed-form expression for the immobile excitation energy as well as Casimir-like forces between immobile excitations due to the dual-chain fragmentation. The results connect fracton-like reduced mobility to exactly solvable 1D spin models and suggest routes to finite-density and finite-temperature behavior and to higher-dimensional generalizations, such as links to XY toric-code-type constructions in 2D.

Abstract

We present a solvable one-dimensional spin-1/2 model on the diamond chain featuring three-spin interactions, which displays both, mobile excitations driving a second-order phase transition between an ordered and a $\mathbb{Z}_2$-symmetry broken phase, as well as non-trivial fully immobile excitations. The model is motivated by the physics of fracton excitations, which only possess mobility in a reduced dimension compared to the full model. We provide an exact mapping of this model to an arbitrary number of independent transverse-field Ising chain segments with open boundary conditions. The number and lengths of these segments correspond directly to the number of immobile excitations and their respective distances from one another. Furthermore, we demonstrate that multiple immobile excitations exhibit Casimir-like forces between them, resulting in a non-trivial spectrum.

Figures (4)

• Figure 1: Diamond lattice with alternating single and dimer sites (blue dots). The three-spin interactions given by products of $\sigma^x$ operators are indicated by the red and blue three-particle bonds labelled $a,b$ respectively. The unit cell composed by sites $A$, $B$, and $C$ is illustrated as the dashed box.
• Figure 2: Dispersion of the three modes of the model at $J/h = 1.2$. The upper mode (red line) is adiabatically connected to two spin-flips on a dimer, the lower mode (blue line) is adiabatically connected to a single spin-flip on a single site. The immobile mode (black line) has a flat dispersion and lies energetically between the two mobile modes but can overlap with the two mobile modes.
• Figure 3: Energy gaps of the lower lying mobile mode (blue line) and the immobile mode (black line) as a function of the parameter $J/h$.
• Figure 4: Force between two immobile excitations separated by a distance of $d$ dimers for different values of $J/h$. Due to the discrete nature of the model the force is calculated as the difference between the energy of two immobile excitations at a distance $d$ and $d-1$.