Altermagnets are a new class of symmetry-compensated magnets with large spin splittings. Here, we show that the notion of altermagnetism extends beyond the realm of Landau-type order: we study exactly solvable Z_{2} quantum spin(-orbital) liquids (QSLs), which simultaneously support magnetic long-range order as well as fractionalization and Z_{2} topological order. Our symmetry analysis reveals that in this model three distinct types of "fractionalized altermagnets (AM^{*})" emerge, which can be distinguished by their residual symmetries. Importantly, the fractionalized excitations of these states carry an emergent Z_{2} gauge charge, which implies that they transform projectively under symmetry operations. Consequently, we show that "altermagnetic spin splittings" are now encoded in a momentum-dependent particle-hole asymmetry of the fermionic parton bands. We discuss consequences for experimental observables such as dynamical spin structure factors and (nonlinear) thermal and spin transport.
Altermagnets are a new class of symmetry-compensated magnets with large spin splittings. Here, we show that the notion of altermagnetism extends beyond the realm of Landau-type order: we study exactly solvable Z_{2} quantum spin(-orbital) liquids (QSLs), which simultaneously support magnetic long-range order as well as fractionalization and Z_{2} topological order. Our symmetry analysis reveals that in this model three distinct types of "fractionalized altermagnets (AM^{*})" emerge, which can be distinguished by their residual symmetries. Importantly, the fractionalized excitations of these states carry an emergent Z_{2} gauge charge, which implies that they transform projectively under symmetry operations. Consequently, we show that "altermagnetic spin splittings" are now encoded in a momentum-dependent particle-hole asymmetry of the fermionic parton bands. We discuss consequences for experimental observables such as dynamical spin structure factors and (nonlinear) thermal and spin transport.
We construct spin- 3 2 and spin- 7 2 models on the square-octagon and checkerboard lattices that are exactly solvable with Majorana representations. They give rise to spin-liquid phases with full spin-rotation and lattice-translational symmetries but broken time-reversal symmetry. Although nonzero on elementary plaquettes, the net orbital magnetic moment is guaranteed to vanish as a result of point symmetries; due to the analogy to long-range ordered altermagnets, these types of phases were dubbed altermagnetic spin liquids in [Sobral ., ]. For the spin- 3 2 model, we find that a g -wave altermagnetic spin liquid emerges as the unique ground state. In contrast, the spin- 7 2 model exhibits a significantly richer phase diagram, involving different types of chiral spin liquids competing with a d -wave altermagnetic spin liquid. Finally, we identify and characterize the topological and nontopological excitations, illustrating the rich physics of altermagnetic spin liquids resulting from the interplay of nontrivial topological and symmetry aspects of this novel phase of matter.
We revisit the fermionic parton approach to S=1/2 quantum spin liquids with SU(2) spin-rotation symmetry, and the associated projective symmetry group (PSG) classification. We point out that the existing PSG classification is incomplete; upon completing it, we find spin-liquid states with S=1 and S=0 Majorana fermion excitations coupled to a deconfined Z2 gauge field. The crucial observation leading us to this result is that, like space group and time-reversal symmetries, spin rotations can act projectively on the fermionic partons; that is, a spin rotation may be realized by simultaneous SU(2) spin and gauge rotations. We show that there are only two realizations of spin rotations acting on fermionic partons: the familiar naive realization where spin rotation is not accompanied by any gauge transformation, and a single type of projective realization. We discuss the PSG classification for states with projective spin rotations. To illustrate these results, we show that there are four such PSGs on the two-dimensional square lattice. We study the properties of the corresponding states, finding that one---with gapless Fermi points---is a stable phase beyond mean-field theory. In this phase, depending on parameters, a small Zeeman magnetic field can open a partial gap for the Majorana fermion excitations. Moreover, there are nearby gapped phases supporting Z2 vortex excitations obeying non-Abelian statistics. We conclude with a discussion of various open issues, including the challenging question of where such S=1 Majorana spin liquids may occur in models and in real systems.
Projective symmetry group classification of fermionic Z2 spin liquids on the dipolar-octupolar pyrochlore magnets
We construct and classify Z2 quantum spin liquids (QSLs) of fermionic partons on the dipolar-octupolar (DO) pyrochlore lattice. The DO pseudospin shares the space-group and time-reversal algebra of the effective spin-1/2 Kramers doublet, its projective symmetry group (PSG) yields the same 48 symmetric Z2 classes as the generic fermionic classification. We show, the distinct DO symmetry action, encoded in a set of sublattice-independent operators, selects a physically different set of symmetry-allowed mean-field channels, so that the resulting ansatze, spinon spectra, and observables differ from both the effective spin-1/2 case and the bosonic DO classification. Decomposing the anisotropic XYZ exchange model into all symmetric singlet and triplet hopping and pairing channels and minimizing the mean-field energy self-consistently, we obtain phase diagrams in the (Jx~,Jz~) plane in which only a few Z2 states, several supporting gapless or flat-band spinon spectra, are stabilized. For these states we compute the spinon band structures, the low-temperature specific heat, and the dynamical spin structure factor, and we argue that the power-law specific-heat exponent is a diagnostic of the spinon nodal structure. A near-linear-in-temperature specific heat together with broad continua in the structure factor gives a fermionic-parton account of the gapless, fractionalized thermodynamics reported in DO materials such as Nd2ScNbO7, complementing the bosonic quantum-spin-ice picture of the Ce-based compounds. The work extends the fermionic-parton classification of DO pyrochlore QSLs beyond the U(1) sector, where Z2 topological order is realized equally naturally by fermionic and bosonic spinons.
Frustrated magnetism provides a rich platform for realizing exotic quantum phases in condensed matter systems. Among these, quantum spin liquids (QSLs) are a particularly intriguing class of states that exhibit no magnetic ordering and preserve all symmetries down to zero temperature. Just as conventional magnets can be thought of as spin crystals,QSLs represent their liquid analogs. This thesis explores two broad types of QSLs in two-dimensional magnets, with a focus on their dynamic properties and the nature of transitions from conventional magnetic states into QSL phases. QSLs are topological phases that host fractionalized excitations known as spinons. In these states, mobile spinons are coupled to an emergent gauge field, forming a highly entangled phase of matter. The first part of this thesis focuses on Dirac spin liquids (DSLs), a class of QSLs where spinons exhibit a gapless Dirac dispersion. To study the dynamicsof these spin liquids beyond leading-order mean-field theory, we develop a novel large-N fermionic limit of gapless spinons. In this limit, all gauge excitations are suppressed, and the leading-order corrections instead arise from order parameter fluctuations, which may condense and drive transitions into symmetry-broken phases from the parent DSL. Incorporating random phase approximation (RPA) corrections to the spin susceptibility, we find that sharp, spin-wave–like modes are recovered in the excitation spectrum. This methodology successfully captures characteristic features of the dynamics near phase transitions and in weakly symmetry-broken phases proximate to a DSL. We apply this approach to microscopic models, including the J1-J2 XXZ model on the triangular lattice and the J1-J3 XXZ model on the honeycomb lattice. These models closely represent a wide range of frustrated materials, such as triangular and honeycomb cobaltates and triangular ytterbium compounds. Our theoretical predictions show excellent agreement with experimentally measured inelastic neutron scattering spectra from these materials. Gapless spinons can also sometimes become gapped through chiral symmetry breaking, leading to the formation of a chiral spin liquid (CSL). The CSL is an archetypal bosonic realization of a fractional quantum Hall (FQH) state at half-filling, and has been proposed as a time-reversal–breaking ground state in a variety of frustrated spin models. The second part of this thesis addresses the intriguing possibility of realizing a CSL through the "melting" of a chiral spin crystal—a conventional magnet with non-coplanar ordering. Such a transition offers a physically appealing analogue to the melting of a Wigner crystal into a fermionic FQH liquid. Focusing on the kagome lattice, where chiral interactions are knownto stabilize a CSL ground state, we study the effects of additional long-range Heisenberg interactions and demonstrate a phase transition into a non-coplanar octahedral magnetic order. To describe the universal class of transition between a CSL and a non-coplanar anti-ferromagnetic order, we then develop a novel Chern-Simons–matter field theory framework, constrained by symmetry principles and symmetry fractionalization patterns. This thesis thus provides new insights into the dynamical signatures and phase transitions of quantum spin liquids, offering a bridge between theoretical models and experimental observations in frustrated magnetic systems.
The presence of multiple competing interactions in spin systems can lead to novel phases known as quantum spin liquids (QSLs). QSLs are disordered spin systems with long-range entanglement that realize the deconfined phase of emergent lattice gauge theories and host fractional excitations. A notable example of great experimental interest is quantum spin ice (QSI), a three-dimensional QSL that realizes an analog of quantum electrodynamics with emergent photons and interacting electric and magnetic charges. The current leading candidate materials for realizing QSI are dipolar-octupolar (DO) cerium-based pyrochlore compounds. These are believed to stabilize a U(1) QSL known as π-flux QSI, where translation symmetry acting on spinon excitations fractionalizes. In this thesis, we explore the physics of QSI, relying heavily on parton constructions, to make experimentally relevant predictions for DO materials. First, we explore the physics of symmetry fractionalization in QSI, whereby lattice symmetries are realized projectively when acting on fractional quasiparticles. Symmetry fractionalization is examined within the framework of gauge mean-field theory (GMFT), a parton construction designed explicitly for application to QSI. This study shows that the pyrochlore lattice space group strongly constrains possible symmetry fractionalization classes. It systematically explains how all space group transformations, beyond translation, fractionalize in π-flux QSI. Next, we apply this framework to DO materials and show how the equal-time structure factor predicted for π-flux QSI is consistent with recent energy-integrated inelastic neutron scattering experiments on certain candidate materials. We also predict that the dynamical spin structure factor of π-flux QSI is characterized by a broad continuum with three distinctive peaks of decreasing intensity that are approximately equally spaced. We argue that these peaks should be accessible with currently available experimental apparatus and offer a smoking-gun signature of π-flux QSI. We confirm those predictions by extending the exclusive boson representation and speculate on the thermodynamic behavior of cerium candidates. We close by commenting on recent experimental breakthroughs and possible future directions.
Quantum spin liquids (QSLs) host a variety of fractionalized particles. In Kitaev's paradigmatic honeycomb model a spin-fractionalizes intoflux due to emergentgauge field and matter Majorana fermions. Although these excitations have well-defined dynamics in the integrable limit, their direct experimental identification is notoriously challenging: realistic materials inevitably host additional symmetry-allowed interactions that break integrability and hybridize gauge and matter sectors, while magnetic fields, which are often required to suppress competing order and stabilize a putative QSL regime, further entangle the responses of different fractionalized quasiparticles and may even drive the system into field-induced spin-liquid phases that are not adiabatically connected to the integrable limit. A prominent example is the quantum Majorana metal, in which the distinct dynamics of fractionalized Majorana fermions can become directly visible in scattering. This review highlights recent progress on these related questions: in which field-stabilized QSL regimes and nearby emergent phases, and under what conditions, can the response of a specific fractionalized quasiparticle be isolated and positively understood, thereby clarifying the existence and the experimental scope of putative spin liquids? We review the progress on these questions across Abelian, non-Abelian, and emergent quantum phases under magnetic field that are not perturbatively connected to the integrable limit. We connect these field-induced dynamical phenomena to concrete experimental observables, relevant for neutron scattering, resonant inelastic x-ray scattering, and pump-probe spectroscopy, techniques that are capable of identifying the nature of different magnetic-field-induced phases and resolving specific types of fractionalized particles, including Majoranas andfluxes.