We design a lattice model of a “mixed” U(1) gauge field coupled to fermions with a flavor chemical potential and solve it with large-scale determinant quantum Monte Carlo simulations. In this mixed construction, dynamical magnetic monopole events are suppressed in the temporal-continuum limit. For zero flavor chemical potential, the model realizes three-dimensional quantum electrodynamics ( QED 3 ) which has been argued to describe the ground state and low-energy excitations of the Dirac spin liquid phase of quantum antiferromagnets. At finite flavor chemical potential, which corresponds to a Zeeman field perturbing the Dirac spin liquid, we find a “chiral flux” phase which is characterized by the generation of a finite mean emergent gauge flux and, accordingly, the formation of relativistic Landau levels for the Dirac fermions. We use field-theory arguments to argue that in this state a U ( 1 ) m magnetic symmetry is spontaneously broken, leading to a gapless photon mode. Because of spin-flux attachment this mode would contribute to the low-energy longitudinal spin response. We numerically compute longitudinal and transverse spin structure factors which match our continuum and lattice mean-field theory predictions. In a different region of the phase diagram, strong fluctuations of the emergent gauge field give rise to an antiferromagnetically ordered state with gapped Dirac fermions coexisting with a deconfined gauge field. We also find an interesting intermediate phase where the chiral flux phase and the antiferromagnetic phase coexist. We argue that our results pave the way to testable predictions for magnetized Dirac spin liquids in frustrated quantum antiferromagnets.
We design a lattice model of a “mixed” U(1) gauge field coupled to fermions with a flavor chemical potential and solve it with large-scale determinant quantum Monte Carlo simulations. In this mixed construction, dynamical magnetic monopole events are suppressed in the temporal-continuum limit. For zero flavor chemical potential, the model realizes three-dimensional quantum electrodynamics ( QED 3 ) which has been argued to describe the ground state and low-energy excitations of the Dirac spin liquid phase of quantum antiferromagnets. At finite flavor chemical potential, which corresponds to a Zeeman field perturbing the Dirac spin liquid, we find a “chiral flux” phase which is characterized by the generation of a finite mean emergent gauge flux and, accordingly, the formation of relativistic Landau levels for the Dirac fermions. We use field-theory arguments to argue that in this state a U ( 1 ) m magnetic symmetry is spontaneously broken, leading to a gapless photon mode. Because of spin-flux attachment this mode would contribute to the low-energy longitudinal spin response. We numerically compute longitudinal and transverse spin structure factors which match our continuum and lattice mean-field theory predictions. In a different region of the phase diagram, strong fluctuations of the emergent gauge field give rise to an antiferromagnetically ordered state with gapped Dirac fermions coexisting with a deconfined gauge field. We also find an interesting intermediate phase where the chiral flux phase and the antiferromagnetic phase coexist. We argue that our results pave the way to testable predictions for magnetized Dirac spin liquids in frustrated quantum antiferromagnets.
Inspired by the recent progress on the Dirac spin liquid and the kagomé lattice antiferromagnets, we revisit the U(1) Dirac spin liquid on the kagomé lattice and consider the response of this quantum state to the weak magnetic field by examining the matter-gauge coupling. Even though the system is in the strong Mott insulating regime, the Zeeman coupling could induce the internal U(1) gauge flux with the assistance of the Dzyaloshinskii-Moriya interaction. In addition to the perturbatively induced nonuniform flux from the microscopic interactions, the system spontaneously generates the uniform U(1) gauge flux in a nonperturbative fashion to create the spinon Landau levels and thus gains the kinetic energy for the spinon matters. Renormalized mean-field theory is employed to validate these two flux generation mechanisms. The resulting state is argued to be an ordered antiferromagnet with the in-plane magnetic order, and the gapless Goldstone mode behaves like the gapless gauge boson and the spinons appear at higher energies. The dynamic properties of this antiferromagnet and the implication for other matter-gauge-coupled systems are discussed.
Theory of Dirac spin liquids on spin-S triangular lattice: possible application to α-CrOOH(D)
Triangular lattice quantum antiferromagnet has recently emerged to be a promising playground for realizing Dirac spin liquids (DSLs) -- a class of highly entangled quantum phases hosting emergent gauge fields and gapless Dirac fermions. While previous theories and experiments focused mainly on S=1/2 spin systems, more recently signals of a DSL were detected in an S=3/2 system α-CrOOH(D). In this work we develop a theory of DSLs on triangular lattice with spin-S moments. We argue that in the most natural scenario, a spin-S system realizes a U(2S) DSL, described at low energy by gapless Dirac fermions coupled with an emergent U(2S) gauge field (also known as U(2S) QCD3). An appealing feature of this scenario is that at sufficiently large S, the U(2S) QCD becomes intrinsically unstable toward spontaneous symmetry breaking and confinement. The confined phase is simply the 120∘ coplanar magnetic order, which agrees with semiclassical (large-S) results on simple Heisenberg-like models. Other scenarios are nevertheless possible, especially at small S when quantum fluctuations are strong. For S=3/2, we argue that a U(1) DSL is also theoretically possible and phenomenologically compatible with existing measurements. One way to distinguish the U(3) DSL from the U(1) DSL is to break time-reversal symmetry, for example by adding a spin chirality term Si⋅(Sj×Sk) in numerical simulations: the U(1) DSL becomes the standard Kalmeyer-Laughlin chiral spin liquid with semion/anti-semion excitation; the U(3) DSL, in contrast, becomes a non-abelian chiral spin liquid described by the SU(2)3 topological order, with Fibonacci-like anyons.
The U(1) Dirac spin liquid might realize an exotic phase of matter whose low-energy properties are described by quantum electrodynamics in 2+1 dimensions, where gapless modes exist but spinons and gauge fields are strongly coupled. Its existence was proposed in frustrated Heisenberg models in the presence of frustrating superexchange interactions by the (Abrikosov) fermionic representation of the spin operators [Wen, Phys. Rev. B 65, 165113 (2002)], supplemented by the Gutzwiller projection. Here, we construct charge-Q monopole excitations in the Heisenberg model on the triangular lattice with nearest-neighbor (J1) and next-nearest-neighbor (J2) couplings. In the highly frustrated regime, singlet and triplet monopoles with Q=1 become gapless in the thermodynamic limit; in addition, the energies for generic Q agree with field-theoretical predictions, obtained for a large number of gapless fermion modes. Finally, we consider localized gauge excitations, in which magnetic $\ensuremath{\pi}$ fluxes are concentrated in the triangular plaquettes (in analogy with Z2 visons), showing that these kinds of states do not play a relevant role at low energies. All our findings lend support to a stable U(1) Dirac spin liquid in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice.
The parton approach for quantum spin liquids gives a transparent description of low-energy elementary excitations, e.g., spinons and emergent gauge-field fluctuations. The latter ones are directly coupled to the hopping/pairing of spinons. By using the fermionic representation of the U(1) Dirac state on the kagome lattice and variational Monte Carlo techniques to include the Gutzwiller projection, we analyze the effect of modifying the gauge fields in the spinon kinematics. In particular, we construct low-energy monopole excitations, which are shown to be gapless in the thermodynamic limit. States with a finite number of monopoles or with a finite density of them are also considered, with different patterns of the gauge fluxes. We show that these chiral states are not stabilized in the Heisenberg model with nearest-neighbor superexchange couplings, and the Dirac state corresponds to the lowest-energy ansatz within this family of variational wave functions. Our results support the idea that spinons with a gapless conical spectrum coexist with gapless monopole excitations, even for the spin-1/2 case.
We study Gutzwiller-projected wave functions for triangular-lattice U(1) Dirac spin liquids in a Zeeman field, where we allow the U(1) gauge field to develop a gauge flux, resulting in (spin-split) spinon Landau levels. We find that at a given magnetization, the optimal candidate state has a finite flux chosen such that the spinon filling lies in a |C|=1 Landau-level gap: it gives the lowest variational energy and the smallest energy variance within our correlation-matrix reconstruction for local Heisenberg-type models. By symmetry, we argue that the finite gauge flux results in a nonzero (staggered) scalar spin chirality, as also numerically observed, and further find that the |C|=1 state exhibits dominant quasi-long-ranged 120° magnetic correlations. Studying the next-to-optimal wave function with a |C|=2 Landau-level gap, we observe unusual spin-nematic correlations. Our results may provide guidance for analyzing the magnetic-field response of Dirac spin liquid candidate materials and offer numerical diagnostics that can connect to the underlying theory of spinons coupled to an emergent U(1) gauge field.
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 (QED3), 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.