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arXiv:2306.12032 · Physical review. B./Physical review. B

Quantum spin liquids on the diamond lattice

Aishwarya Chauhan, Atanu Maity, Chunxiao Liu, Jonas Sonnenschein, Francesco Ferrari, Yasir Iqbal

Published Oct 18, 2023• 10 citations
arXivHTMLPDF

Abstract

The diamond lattice has been a promising playground for both classical and quantum spin liquids. Here, the authors conduct a comprehensive symmetry classification of $S$=\textonehalf{} quantum spin liquids with SU(2), U(1), and \ensuremath{\mathbb{Z}}${}_{2}$ gauge fields, and present dynamical spin structure factors within a self-consistent simulation. The authors highlight three discoveries: spinon Ans\"atze with robust gapless nodal loops, topological bands realizing topological insulators, and that Gutzwiller projection of the 0- and $\ensuremath{\pi}$-flux SU(2) spin liquids generates long-range N\'eel order.

DOI
10.1103/physrevb.108.134424

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The diamond lattice has been a promising playground for both classical and quantum spin liquids. Here, the authors conduct a comprehensive symmetry classification of SSS=\textonehalf{} quantum spin liquids with SU(2), U(1), and \ensuremath{\mathbb{Z}}2{}_{2}2​ gauge fields, and present dynamical spin structure factors within a self-consistent simulation. The authors highlight three discoveries: spinon Ans"atze with robust gapless nodal loops, topological bands realizing topological insulators, and that Gutzwiller projection of the 0- and \ensuremathπ\ensuremath{\pi}\ensuremathπ-flux SU(2) spin liquids generates long-range N'eel order.
We study fermionic quantum spin liquids (QSLs) on the three-dimensional trillium lattice of corner-sharing triangles. We are motivated by recent experimental and theoretical investigations that have explored various classical and quantum spin liquid states on similar networks of triangular motifs with strong geometric frustration. Using the framework of projective symmetry groups (PSG), we obtain a classification of all symmetric Z 2 and U ( 1 ) QSLs on the trillium lattice. We find two Z 2 spin-liquids, and a single U ( 1 ) spin-liquid that is proximate to one of the Z 2 states. The small number of solutions reflects the constraints imposed by the nonsymmorphic symmetries in the space group of the trillium lattice. Using self-consistency conditions of the mean-field equations, we obtain the spinon band-structure and spin structure factors corresponding to these states. All three of our spin liquids are gapless at their saddle points: one of the two Z 2 QSLs is nodal, while the U ( 1 ) case hosts a spinon Fermi surface. One of our Z 2 spin liquids hosts a stable gapless nodal star that is protected by projective symmetries against additions of further neighbor terms in the mean-field ansatz. We comment on directions for further work.
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Motivated by recent numerical studies reporting putative quantum paramagnetic behavior in spin-1/21/21/2 Heisenberg models on the maple-leaf lattice, we classify Abrikosov fermion mean-field Ans\"atze of fully symmetric U(1) and Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquids within the framework of projective symmetry groups. We obtain a total of 17U(1)17\phantom{\rule{4pt}{0ex}}\text{U}(1)17U(1) and 12Z212\phantom{\rule{4pt}{0ex}}{\mathbb{Z}}_{2}12Z2​ algebraic PSGs, and, upon restricting their realization via mean-field Ans\"atze with nearest-neighbor amplitudes (relevant to the studied models), only 12 U(1) and 8 Z2{\mathbb{Z}}_{2}Z2​ distinct phases are obtained. We present both singlet and triplet fields for all Ans\"atze up to third nearest-neighbor bonds and discuss their spinon dispersions as well as their dynamical spin structure factors. We further assess the effects of Gutzwiller projection on the equal-time spin structure factors.
Spin liquids represent exotic types of quantum matter that evade conventional symmetry-breaking order even at zero temperature. Exhaustive classifications of spin liquids have been carried out in several systems, particularly in the presence of full SU(2) spin-rotation symmetry. Real magnetic compounds, however, generically break SU(2) spin symmetry as a result of spin-orbit coupling---which in many materials provides an ``order one'' effect. We generalize previous works by using the projective symmetry group method to classify Z2{\mathbb{Z}}_{2}Z2​ spin liquids on the square lattice when SU(2) spin symmetry is maximally lifted. We find that, counterintuitively, the lifting of spin symmetry actually results in vastly more spin-liquid phases compared to SU(2)-invariant systems. A generic feature of the SU(2)-broken case is that the spinons naturally undergo p+ipp+ipp+ip pairing; consequently, many of these Z2{\mathbb{Z}}_{2}Z2​ spin liquids feature a topologically nontrivial spinon band structure supporting gapless Majorana edge states. We study in detail several spin-liquid phases with varying numbers of gapless edge states and discuss their topological protection. The edge states are often protected by a combination of time reversal and lattice symmetries and hence resemble recently proposed topological crystalline superconductors.
Quantum spin liquids are novel phases of matter whose physical properties cannot be captured within the conventional framework of phase transitions. These quantum states are characterized by excitations, called spinons, carrying a fractional value of the integer spin. A description based on a mean-field approximation leads to effective gauge theory. Due to this gauge freedom symmetry group operations act via projective symmetry group (PSG) representations, which can be used to classify quantum states beyond the conventional method. This introducing discussion is found in Chapters 2-6. A preceding PSG classification on the square lattice, which took spin rotational symmetry breaking terms into account, showed that the resulting mean-field states could exhibit topologically non-trivial spinon bands. In Chapter 7, only those el- ements of this Classification are studied, which lead to short-ranged mean-field models. The mutual effect of spinons and visons is scrutinized by adding static configurations of a background gauge field. Spinons and visons form composite ob- jects, which in the case of topological spinon bands, are described by Majorana zero modes. An exact mapping corroborates the relation to topological supercon- ductors and vortices. The effect of multiple visons is investigated, and a possible experimental signature is exposed. Chapter 8 explains how spinon mean-field theory can be used to derive a phe- nomenological effective model for the compound Ca 10 Cr 7 O 28 . A material that was recently proposed as a quantum spin liquid. This low-energy theory consists of spinon hopping processes. The underlying bilayer kagome structure can be simpli- fied to an effective honeycomb lattice. The dynamical spin structure factor is calcu- lated and compared to neutron scattering data. It is discussed how f -wave pairing of spinons can explain small deviations from a linear temperature dependency observed in heat capacity measurements at the low temperatures. This pairing mechanism breaks the U (1) invariance group of the pure hopping model and renders a coherent picture of Ca 10 Cr 7 O 28 being a Z 2 quantum spin liquid. Chapter 9 carries out a PSG classification for the simple-, body centered-, and face centered cubic lattices. Despite a large number of possible PSG representations, only very few describe short-ranged mean-field states. These mean-field models are constructed up to third neighbor couplings and calculated in a self-consistent manner. It is found that the energy is lower for non-trivial sign patterns induced by non-trivial representations of the symmetry generators. Higher neighbor extensions, including pairing terms, yield lower mean-field ground state energies compared to extensions with only hopping terms. For the fcc lattice, a first neighbor state could be identified, which exhibits symmetry protected zero-energy modes.
Despite rapid progress in understanding gapped topological states, much less is known about gapless topological phases of matter, especially in strongly correlated electrons. In this work, we discuss a large class of robust gapless quantum spin liquids in frustrated magnets made of half-integer spins, which are described by gapless fermionic spinons coupled to dynamical Z2{\mathbb{Z}}_{2}Z2​ gauge fields. Requiring U(1)\text{U}(1)U(1) spin conservation, time-reversal, and certain space-group symmetries, we show that certain spinon symmetry fractionalization class necessarily leads to a gapless spectrum. These gapless excitations are stable against any perturbations, as long as the required symmetries are preserved. Applying these gapless criteria to spin-12\frac{1}{2}21​ systems on square, triangular, and kagome lattices, we show that all gapped symmetric Z2{\mathbb{Z}}_{2}Z2​ spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.