Researchers have identified 50 U(1) and 64 projective symmetry groups (PSGs) potentially existing within the ruby lattice model, revealing a surprisingly complex array of possible quantum states. Atanu Maity of Julius-Maximilians-Universität Würzburg and the Indian Institute of Technology Madras, along with colleagues, found that focusing on realistic physical interactions, specifically, those extending to second-nearest neighbors, significantly narrowed the possibilities to just 8 U(1) and 18 distinct states.

This work builds on growing interest in the ruby lattice, recently realized in experiments using Rydberg atom arrays to create topologically ordered quantum spin liquids. The team’s analysis provides a detailed map of potential quantum phases within this increasingly important system.

Ruby Lattice as a Platform for Quantum Spin Liquids

This extensive catalog suggests a far richer landscape of possibilities than previously understood for this increasingly studied system, hinting at subtle distinctions in how electron spins might arrange themselves without conventional magnetic order. Restricting their analysis to mean-field approaches incorporating interactions between neighboring spins up to the second-nearest neighbor dramatically reduced the number of viable states to just 8 U(1) structures and 18 distinct overall configurations.

This reduction demonstrates that not all theoretically possible arrangements are stabilized by the kinds of interactions expected in real materials, providing a filter for future investigations. These arrays allow precise control over interactions between atoms, mimicking the frustrated magnetic interactions needed to stabilize spin liquid states. The researchers also examined the impact of longer-range interactions, extending their analysis to include third-nearest-neighbor bonds to understand how these couplings might influence the resulting spinon dispersions and dynamical spin structure factors.

Projective Symmetry Group Classification of U(1) and Fermionic Spinons

Numerical studies of ruby-lattice spin models reveal evidence of both gapped and gapless spin-liquid ground states, each exhibiting distinct low-energy gauge structures. These findings motivated a systematic classification of potential quantum phases. Researchers employed projective symmetry groups to analyze U(1) and fermionic spinon mean-field theories, specifically focusing on spin-1/2 systems while respecting space-group, time-reversal and spin-rotation symmetries.

A key concept within the framework is the invariant gauge group, defined as the symmetry group of the ground-state mean-field Ansatz, and representing the low-energy symmetry group of the system. The identity element within any symmetry group is defined up to an element of this gauge group, an important aspect of the projective symmetry group formalism.

Researchers constructed quadratic spinon Hamiltonians on the ruby lattice, classifying them with U(1) and $Z_{2}$ low-energy gauge groups, and this work builds on the established success of this method in describing ground states and low-energy behaviors of quantum spin models on two- and three-dimensional lattices. The resulting catalog of potential phases offers a detailed map for future investigations into the complex behavior of quantum spin liquids.

Mean-Field Ansätze and Realizable QSL States on the Ruby Lattice

Classifying mean-field theories for quantum spin liquids reveals that of 50 potential U(1) states on the ruby lattice, only 8 are stabilized by interactions limited to second-nearest neighbors. The analysis extends to 64 distinct projective symmetry groups initially, but realistic interactions drastically reduce the number of viable arrangements. The team’s work builds on the understanding that the ruby lattice, a geometrically frustrated structure, is particularly conducive to hosting these exotic quantum states.

Specifically, the study examined mean-field Ansätze, approximations used to simplify complex quantum many-body problems, with singlet amplitudes extending up to third-nearest-neighbor bonds. Extending the range of these interactions to include third neighbors increases the number of realizable U(1) states to 18, from the initial 8 found with only first- and second-neighbor interactions.

The researchers also investigated 64 $Z_{2}$ states, finding that 22 can be realized with similar mean-field approaches. The properties of each Ansatz, and its resulting excitations, are determined by a set of mean-field parameters.

These parameters, the researchers note, “should be computed self-consistently by optimization with respect to a given model.” The team focused on models resembling the isotropic S = 1/2 Heisenberg antiferromagnet on the ruby lattice, where the first and second neighbor couplings are equal and third-neighbor coupling is zero. This focus reflects evidence from tensor-network results suggesting a gapless quantum spin liquid ground state in such systems.

Singlet Amplitudes and Spinon Dispersions for QSL Analysis

Numerical studies reveal a diversity of potential quantum spin liquid ground states on the ruby lattice, exhibiting both gapped and gapless low-energy gauge structures, motivating a detailed classification of possible arrangements. Spinon band structures for the eight U(1) Ansätze realizable with first- and second-nearest-neighbor mean-field amplitudes were calculated, with the magnitude of symmetry-allowed hoppings set to one for $χ_{1}$ = $χ_{1}^{′}$ , and zero for further-neighbor bonds.

Analysis of the dynamical spin structure factor (DSF) revealed finite intensity down to zero energy in some cases, indicative of a Fermi surface. Specifically, the U30ππ Ansatz consistently exhibited a gapped spectrum with high intensity around the $M^{′}$ point at approximately 4.3 energy units, while the U3π0π Ansatz also showed generically gapped spinon dispersions reflected in the DSF.

Phase Diagram of the J_1-J_1′-J_2 Heisenberg Model

The resulting phase diagram was constructed using a self-consistent mean-field approximation, revealing potential regions hosting distinct quantum spin liquid (QSL) phases. The analysis identified extended regions within the parameter space supporting two separate QSL phases, suggesting a richer variety of ground states than previously understood for this lattice geometry. The ruby lattice’s inherent frustration, arising from its arrangement of antiferromagnetically interacting spins, is a key factor in enabling these QSL phases.

Recent experiments using Rydberg atom arrays have successfully prepared a topologically ordered Z_2 QSL on this lattice, corresponding to a resonating valence bond (RVB) phase of an underlying quantum dimer model. The connection between phases on the ruby lattice and the anisotropic Kitaev honeycomb model further expands the potential for cross-validation and deeper understanding of these exotic quantum states.

This work builds on the understanding that lattice geometries inducing frustration are highly desirable for realizing enigmatic quantum spin liquids. The identification of multiple QSL phases within the J_1-J_1’-J_2 model, and the detailed mapping of their spinon band structures and dynamical spin structure factors, represent a step toward characterizing these complex systems and potentially harnessing their unique properties. The authors obtain the phase diagram of the Heisenberg model in the presence of first (
J_1), second (
J_1^′), and third (
J_2) neighbor antiferromagnetic couplings within a self-consistent mean-field approximation.

Connections Between Ruby and Maple-Leaf Lattice QSLs

This suggests that longer-range interactions play a role in stabilizing specific QSL phases on the ruby lattice. The study’s focus on singlet QSLs aligns with tensor-network results indicating a gapless QSL state within the Heisenberg model with purely antiferromagnetic couplings, providing a valuable connection between theoretical predictions and numerical observations. The research also illuminates a potential relationship between the ruby and maple-leaf lattices, suggesting that bond depletion, reducing the number of connections between spins, could drive a quantum phase transition.

Specifically, the team proposes that a gapped QSL on the maple-leaf lattice may transition to a gapless QSL on the ruby lattice as bonds are removed. Future work could assess these mean-field Ansätze within the pseudofermion functional renormalization group framework, using effective low-energy vertex functions to compute theories for emergent spinon excitations and refine the understanding of these complex quantum phases.

Experimental Realization of a Z_2 Quantum Spin Liquid

Simulations reveal the ruby lattice, a structure recently realized in Rydberg atom arrays, potentially supports a surprisingly diverse array of quantum spin liquid phases. This abundance of potential arrangements suggests the ruby lattice’s frustrated interactions allow for a wide range of topological order. The team presented the resulting singlet fields for all these arrangements, extending their analysis to include interactions spanning up to third-nearest-neighboring bonds to fully map the phase diagram.

Analysis of these arrangements revealed key features of the potential quantum spin liquids. The spectrum of one particular ansatz, labeled U1πππ, exhibits nodal structures at the Fermi level under specific parameter conditions, while transitioning to a Dirac spin liquid when those parameters shift.

Bi14Rh3I9 as a Solid-State Ruby Lattice System

The layered material Bi14Rh3I9 provides a physical system for realizing the ruby lattice structure, offering a potential platform to test theoretical predictions about quantum spin liquids. This realization is significant because it connects abstract models of frustrated magnetism to actual materials, an important step in validating and exploring these exotic quantum states. The ruby lattice, previously demonstrated in Rydberg atom arrays, now has a solid-state analogue, opening avenues for complementary experimental investigation.

This generalization allows for inequivalent interactions between nearest neighbors, modeled by the Hamiltonian H = + +. The researchers note that the internal gauge group (IGG) differs between the ruby lattice and its fermionic representation, precluding a SU(2) IGG and limiting possibilities to U(1) or $Z_{2}$ groups. The team constructed all possible gauge-inequivalent representations for these groups, identifying distinct quantum spin liquid states differentiated by the projective action of symmetry operations. This detailed mapping of symmetry-allowed states provides a framework for identifying these phases through spectroscopic signatures, furthering the search for novel quantum materials.

👉 More information
🗞 Gapped and gapless quantum spin liquids on the ruby lattice
✍️ Atanu Maity, Rhine Samajdar and Yasir Iqbal
🧠 DOI: http://link.aps.org/doi/10.1103/h3d9-mmx2

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