The interplay of symmetry and topology has been at the forefront of recent progress in quantum matter.Here, we uncover an unexpected connection between band topology and the description of competing orders in a quantum magnet.Specifically, we show that aspects of band topology protected by crystalline symmetries determine key properties of the Dirac spin liquid (DSL), which can be defined on the honeycomb, square, triangular, and kagome lattices.At low energies, the DSL on all of these lattices is described by an emergent quantum electrodynamics (QED 3 ) with N f ¼ 4 flavors of Dirac fermions coupled to a Uð1Þ gauge field.However, the symmetry properties of the magnetic monopoles, an important class of critical degrees of freedom, behave very differently on different lattices.In particular, we show that the lattice momentum and angular momentum of monopoles can be determined from the charge (or Wannier) centers of the corresponding spinon insulator.We also show that for DSLs on bipartite lattices, there always exists a monopole that transforms trivially under all microscopic symmetries owing to the existence of a parent SUð2Þ gauge theory.We connect our results to generalized Lieb-Schultz-Mattis theorems and also derive the time-reversal and reflection properties of monopoles.Our results indicate that recent insights into free-fermion band topology can also guide the description of strongly correlated quantum matter.
Abstract Quantum magnets provide the simplest example of strongly interacting quantum matter, yet they continue to resist a comprehensive understanding above one spatial dimension. We explore a promising framework in two dimensions, the Dirac spin liquid (DSL) — quantum electrodynamics (QED 3 ) with 4 Dirac fermions coupled to photons. Importantly, its excitations include magnetic monopoles that drive confinement. We address previously open key questions — the symmetry actions on monopoles on square, honeycomb, triangular and kagome lattices. The stability of the DSL is enhanced on triangular and kagome lattices compared to bipartite (square and honeycomb) lattices. We obtain the universal signatures of the DSL on triangular and kagome lattices, including those of monopole excitations, as a guide to numerics and experiments on existing materials. Even when unstable, the DSL helps unify and organize the plethora of ordered phases in correlated two-dimensional materials.
Dirac spin liquids represent a class of highly-entangled quantum phases in two dimensional Mott insulators, featuring exotic properties such as critical correlation functions and absence of well-defined low energy quasi-particles. Existing numerical works suggest that the spin-orbital
SU(4) symmetric Kugel-Khomskii model of Mott insulators on the honeycomb lattice realizes a Dirac spin-orbital liquid, described at low energy by
(2+1)d quantum electrodynamics (QED
3) with
Nf=8 Dirac fermions. We generalize methods previously developed for
SU(2) spin systems to analyze the symmetry properties and stability of the Dirac spin-orbital liquid. We conclude that the standard Dirac state in the
SU(4) honeycomb system, based on a simple parton mean-field ansatz, is intrinsically unstable at low energy due to the existence of a monopole perturbation that is allowed by physical symmetries and relevant under renormalization group flow. We propose two plausible alternative scenarios compatible with existing numerics. In the first scenario, we propose an alternative
U(1) Dirac spin-orbital liquid, which is similar to the standard one except for its monopole symmetry quantum numbers. This alternative
U(1) state represents a stable gapless phase. In the second scenario, we start from the standard
U(1) Dirac liquid and Higgs the
U(1) gauge symmetry down to
Z4. The resulting
Z4 Dirac spin-orbital liquid is stable. We also discuss the continuous quantum phase transitions from the
Z4 Dirac liquids to conventional symmetry-breaking orders, described by the QED
3 theory with
Nf=8 supplemented with a critical charge-
4 Higgs field. We discuss possible ways to distinguish these scenarios in numerics. We also extend previous calculations of the quantum anomalies of QED
3 and match with generalized lattice Lieb-Schultz-Mattis constraints.
The Dirac spin liquid (DSL) is a two-dimensional (2D) fractionalized Mott insulator featuring massless Dirac spinon excitations coupled to a compact U(1) gauge field, which allows for flux-tunneling instanton events described by magnetic monopoles in (2+1)D Euclidean spacetime. The state-operator correspondence of conformal field theory has been used recently to define associated monopole operators and determine their quantum numbers, which encode the microscopic symmetries of conventional ordered phases proximate to the DSL. In this work, we utilize semiclassical instanton methods not relying on conformal invariance to construct monopole operators directly in (2+1)D spacetime as instanton-induced ’t Hooft vertices, i.e., fermion-number-violating effective interactions originating from zero modes of the Euclidean Dirac operator in an instanton background. In the presence of a flavor-adjoint fermion mass, resummation of the instanton gas is shown to select the correct monopole to be proliferated, in accordance with predictions of the state-operator correspondence. We also show that our instanton-based approach is able to determine monopole quantum numbers on bipartite lattices.
The
U(1) Dirac spin liquid provides a useful organizing framework for frustrated magnets: it offers an algebraic parent state from which competing orders, confinement patterns, and low-energy spectral features can be understood. Whether such a state can occur as a stable ground state of a two-dimensional spin Hamiltonian remains an open question, because monopole operators of the compact gauge field can proliferate and confine the spinons. Here we determine the symmetry quantum numbers of the fundamental monopoles of a maple-leaf-lattice Dirac spin liquid whose generic six-node regime realizes QED
3 with
Nf=12 two-component Dirac fermions. The charge-one monopoles span a
924-dimensional complex representation. In the spin-singlet, zero-momentum, zero-
C6-angular-momentum sector we find five complex charge-one directions, of which three are mapped to their Hermitian conjugates under time reversal. Their real and imaginary Hermitian combinations therefore supply six independent monopole perturbations that are trivial under translations,
C6, time reversal, and spin rotation. The Dirac spin liquid phase is thus not protected by symmetry exclusion of charge-one monopoles: its stability depends on whether these allowed perturbations are relevant in the renormalization group sense. Existing large-
Nf and numerical estimates of the charge-one monopole scaling dimension lie close to the relevance threshold, making the maple-leaf lattice a concrete platform for testing the stability of compact QED
3 in a quantum magnet. Our complete monopole classification also identifies the spin, momentum, and rotation sectors in which monopole excitations should appear, providing symmetry-resolved benchmarks for exact-diagonalization and variational tests of the
Nf=12 Dirac spin liquid.
Quantum spin liquids are exotic phases of quantum matter especially pertinent to many modern condensed matter systems. Dirac spin liquids (DSLs) are a class of gapless quantum spin liquids that do not have a quasi-particle description and are potentially realized in a wide variety of spin 1/2 1 / 2 magnetic systems on 2d 2 d lattices. In particular, the DSL in square lattice spin- 1/2 1 / 2 magnets is described at low energies by (2+1)d ( 2 + 1 ) d quantum electrodynamics with N_f=4 N f = 4 flavors of massless Dirac fermions minimally coupled to an emergent U(1) U ( 1 ) gauge field. The existence of a relevant, symmetry-allowed monopole perturbation renders the DSL on the square lattice intrinsically unstable. We argue that the DSL describes a stable continuous phase transition within the familiar Neel phase (or within the Valence Bond Solid (VBS) phase). In other words, the DSL is an ``unnecessary’’ quantum critical point within a single phase of matter. Our result offers a novel view of the square lattice DSL in that the critical spin liquid can exist within either the Neel or VBS state itself, and does not require leaving these conventional states.
A complicating factor in the realization and observation of quantum spin liquids in materials is the ubiquitous presence of other degrees of freedom, in particular lattice distortion modes (phonons). These provide additional routes for relieving magnetic frustration, thereby possibly destabilizing spin-liquid ground states. In this work, we focus on triangular-lattice Heisenberg antiferromagnets, where recent numerical evidence suggests the presence of an extended U(1) Dirac spin liquid phase which is described by compact quantum electrodynamics in 2+1 dimensions (QED
3), featuring gapless spinons and monopoles as gauge excitations, and believed to flow to a strongly-coupled fixed point with conformal symmetry. Using complementary perturbation theory and scaling arguments, we show that a symmetry-allowed coupling between (classical) finite-wavevector lattice distortions and monopole operators of the U(1) Dirac spin liquid generally induces a spin-Peierls instability towards a (confining) 12-site valence-bond solid state. We support our theoretical analysis with state-of-the-art density matrix renormalization group simulations. Away from the limit of static distortions, we demonstrate that the phonon energy gap establishes a parameter regime where the spin liquid is expected to be stable, and show that the monopole-lattice coupling leads to softening of the phonon in analogy to the Kohn anomaly. We discuss the applicability of our results to similar systems, in particular the Dirac spin liquid on the Kagome lattice.