Pinch-line spin liquids as layered Coulomb phases and applications to cubic models

TL;DR

This work provides a generic, designable framework to create pinch-line spin liquids by stacking 2D algebraic spin liquids into 3D structures, transforming point-like pinch features into 1D lines in reciprocal space. The approach relies on constrainers and critical vectors to realize Gauss-law-like tensor fields, with two main recipes yielding either a single layered Coulomb phase or coexisting, interfering Coulomb sectors along the pinch line. The authors demonstrate concrete realizations on Kagome, snowflake, and checkerboard lattices, revealing how interlayer couplings and intermediate sublattices tune the evolution of pinch points along the line and enhance zero modes, as validated by Monte Carlo simulations and Henley’s projective method. They extend the framework to non-layered, cubic-symmetric lattices (octahedral and octochlore), showing pinch lines can form along multiple equivalent directions and can involve higher-rank tensor fields, potentially hosting fracton-like excitations. Overall, the paper provides a versatile, high-capability platform for designing and understanding pinch-line spin liquids with broad implications for frustrated magnetism and emergent gauge theories, including prospects for experimental realization and exploration of quantum regimes.

Abstract

Spin liquids form fluctuating magnetic textures which have to obey certain rules imposed by frustration. These rules can often be written in the form of a Gauss law, indicating the local conservation of an emergent electric field. In reciprocal space, these emergent Gauss laws appear as singularities known as pinch points, that are accessible to neutron-scattering measurements. But more exotic forms of electromagnetism have been stabilized in spin liquids, and in a few rare instances, these zero-dimensional singularities have been extended into one-dimensional pinch lines. Here we propose a simple framework for the design of pinch-line spin liquids in a layered structure of two-dimensional algebraic spin liquids. A plethora of models can be build within this framework, as exemplified by several concrete examples where our theory is confirmed by simulations, and where the rank of the tensorial gauge field is continuously varied along the pinch line, opening new avenues in fractonic matter. Then we use our framework to understand how the evolution of the singularity pinch point along the pinch line can be understood as the interference pattern of two emergent electric fields. Finally, we apply our intuition on these emergent electric fields in real space to generic pinch line models beyond our layered framework, and revisit the recently proposed pinch line model on the octochlore lattice.

Figures (16)

• Figure 1: a) Illustration of the pinch-line framework. Planes containing a parent 2D algebraic spin liquid hosting pinch points (depicted as a checkerboard) are placed regularly along a transverse axis, with intermediate layers (in blue) designed to make the pinch points one-dimensional. b) Representation of a pinch line in reciprocal space (black line), with pinch points centered on the line on every transverse plane.
• Figure 2: First recipe: Schematic representation of the simple approach to build a spin system hosting a pinch line, presented in section \ref{['sec:simple']}. First, consider a 2D parent model known to host pinch points; here the kagome antiferromagnet with two types of triangular clusters (pink and blue). Next, build a 3D stacking of this 2D parent model with consecutive layers separated by distance $d$. Finally extend the 2D clusters to 3D by linking together two clusters from two consecutive layers.
• Figure 3: Second recipe: (a) Kagome lattice composed of three sublattices, shown in different colors. The lattice can be described as a set of connected clusters, with up and down triangles highlighted in light blue and light magenta, respectively. (b) Projection of the 3D model: the fourth sublattice (green squares) is placed above and below each hexagon of the parent kagome lattice.
• Figure 4: Equal-time structure factor for the 3D generalization of kagome lattice presented in Fig. \ref{['fig: 3D kagome lattice']}(b). The rows show the evolution in the $(q_x,q_y)$ plane orthogonal to the pinch lines (in $a^{-1}$ units). For each panel, the right side is obtained analytically using Henley's projective method at zero temperature Henley2005 [Appendix \ref{['Appendix B: Definition of the structure factor']}], while the left side results from Monte Carlo simulations at very low temperatures $T/J_\text{max} = 2\times 10^{-4}$ [Appendix \ref{['Appendix C: Monte Carlo simulations']}]. Keeping in mind the difference of temperature between the two methods, the agreement between simulations and theory is excellent; in particular the position and persistence of the pinch-point singularities as a function of $q_z$.
• Figure 5: The left panels depict the local cluster structure of the two systems studied here. The plain and crossed circles depict the two sublattices of the 2D parent systems. Blue, red and yellow circles respectively appear with coefficient 1, $\gamma_1$ ($\gamma$ for the snowflake) and $\gamma_2$ in the constrainer C. The squares represent the third sublattice located in intermediate planes; locally, they form square (top) and hexagonal (bottom) bipyramids with the 2D parent lattice. The green and pink squares appear with coefficient $\delta_1$ ($\delta$ for the checkerboard) and $\delta_2$ in the constrainer $\bm{\mathcal{C}}$. The three right columns show the evolution of the equal-time structure factor in $(q_x,q_y)$ planes orthogonal to the pinch lines (in $a^{-1}$ units), with $\gamma = 1/2, \delta_1 = 1$ for the 3D snowflake and $\gamma_1 = 1, \gamma_2 = -1/3$ and $\delta = 1$ for the 3D checkerboard model. For each panel, the right side is obtained analytically using Henley's projective method at zero temperature Henley2005 [Appendix B], while the left side results from Monte Carlo simulations at very low temperatures [Appendix C]. Keeping in mind the difference of temperature between the two methods, the agreement between simulations and theory is excellent, in particular the position and persistence of the pinch-point singularities as a function of $q_z$.