The standard model of particle physics, the most successful description of our universe is formulated in terms of gauge theories. Elementary particles and the excitations of the gauge fields that mediate interactions between them make up the standard model. While gauge theories are fundamental in particle physics and require high energy probes (like a collider) to study them, they may emerge naturally in certain condensed matter systems. In these systems, emergent gauge fields describe the low energy physics and can be more easily studied in a laboratory. Quite often, they are intimately linked to the quantum entanglement and topological properties of the many-particle system at hand. These connections are routinely manifested by excitations that are fractional. They are not bound to each other but can only be created or destroyed locally in pairs and carry fractions of an electronic quantum number.
A quintessential example is a quantum spin liquid (QSL). While the elementary excitations of a magnetically ordered system (e.g. anti-ferromagnet) are magnons with spin quantum number 1, the quasiparticles of a QSL may carry only a fraction of this quantum number. A celebrated model for a QSL is the Kitaev honeycomb model, a two-dimensional (2D) honeycomb lattice of interacting spins. In this model, a local spin flip fractionalizes into a pair of emergent gauge field excitations – visons , and Majorana fermions which represent half of an ordinary fermion. Surprisingly, there is a large class of materials believed to be described by the Kitaev model with additional perturbations— known as Kitaev materials. Understanding how these perturbations influence the nature and signatures of the fractional quasiparticles is crucial for detecting and controlling them in real materials. One major effect of these perturbations is that they make the visons dynamical degrees of freedom. Despite some experimental evidence suggesting the presence of a spin liquid in certain Kitaev materials, strong sample dependence has hindered the unambiguous detection of an emergent dynamical gauge field.
Three different stacking arrangements of Kitaev honeycomb spin liquids. Each layer (shown in varying opacity) is a Kitaev model with spin-1/2 particles on the sites. The spins that sit directly on top (or bottom) of each other are coupled by a Heisenberg exchange.
An often-overlooked aspect of candidate materials is their three-dimensional (3D) structure composed of weakly interacting 2D layers. It is important to understand how the quasiparticles of the purely 2D model behave when a third spatial dimension (stacking axis) is introduced. Our work addresses this question in detail by analyzing the dynamics of the emergent gauge field in simplified models of layer-stacked Kitaev spin liquids.
The 2D honeycomb layers can be stacked in various configurations, much like assembling a structure with Lego bricks. We considered three main types of stacking arrangements commonly seen in real material. The simplest arrangement is called AA stacking, where each layer lies directly on top of another . Other stacking patterns, like the AB and ABC arrangements, where adjacent layers are shifted relative to each other in the layer-plane are more common in materials. We demonstrate that a rich interplay of fractionalization and conservation laws impose severe constraints on the dynamics of visons in these models.
Dynamical excitations of a multilayer Kitaev model. Each shaded hexagon represents a single vison, the elementary excitation of the emergent gauge field which is immobile when alone. The arrows signifies the hopping of a vison pair (from light shaded hexagons to dark shaded ones). Interlayer vison pairs made of visons from two adjacent layers are confined to move within the layer planes while intralayer pairs can move across layers.
Topology prevents a single vison from being created or annihilated locally within a layer, thus blocking the motion of individual visons between layers. Conservation laws further restrict their in-plane motion, rendering single visons completely immobile. When the layers are AA stacked, the parity of visons (odd or even number) in a given hexagonal column is conserved. For AB and ABC stacking, however, conservation laws take the form of 2D “sheets” that criss-cross the system. Single visons cannot cross a sheet, thus are effectively trapped by the sheets. Using these principles, we show that visons form pairs – interlayer and intralayer pairs, which become mobile due to the weak interlayer coupling. While an interlayer pair is constrained to move parallel to the layers, an intralayer pair can only move between layers. Remarkably, the sheet conservation laws put further constraints on their motion. While in an AA stacked model, interlayer pairs have a fully 2D motion, they are constrained by the sheets to move along 1D channels in AB and ABC models. We further calculate the dispersion relations of these excitations and describe how spatial anisotropy in spin-spin interactions as well as magnetic fields profoundly affect their propagation.
This unusual mix of excitations with different mobility constraints in a rather simple model may be used as a starting point to predict novel signatures of fractional quasiparticles in Kitaev materials. For example, one could use a laser pulse to create a high density of excitations at the top surface of a sample and track using optical measurements how they diffuse in space as determined by their dimensionality. Thus, this approach transforms the layered nature of spin liquid materials from a perceived drawback into a valuable tool for exploring fractionalization in real materials.